Taxi, 4430 bytes

Go to Post Office:w 1 l 1 r 1 l.
Pickup a passenger going to The Babelfishery.
Go to The Babelfishery:s 1 l 1 r.
Pickup a passenger going to Cyclone.
0 is waiting at Starchild Numerology.
0 is waiting at Starchild Numerology.
Go to Starchild Numerology:n 1 l 1 l 1 l 2 l.
Pickup a passenger going to Rob's Rest.
Pickup a passenger going to Addition Alley.
Go to Rob's Rest:w 1 r 2 l 1 r.
Go to Cyclone:s 1 l 1 l 2 l.
[B]
Pickup a passenger going to Divide and Conquer.
1 is waiting at Starchild Numerology.
Go to Starchild Numerology:n 1 r 3 l.
Pickup a passenger going to Addition Alley.
Go to Addition Alley:w 1 r 3 r 1 r 1 r.
Pickup a passenger going to Cyclone.
Go to Cyclone:n 1 l 1 l.
Pickup a passenger going to Sunny Skies Park.
Pickup a passenger going to Divide and Conquer.
Go to Divide and Conquer:n 2 r 2 r 1 r.
Pickup a passenger going to Cyclone.
Go to Sunny Skies Park:e 1 l 1 l 2 l 1 l.
Go to Zoom Zoom:n 1 r.
Go to Cyclone:w.
Pickup a passenger going to Sunny Skies Park.
Pickup a passenger going to Equal's Corner.
Pickup a passenger going to Trunkers.
Go to Sunny Skies Park:n 1 r.
Go to Trunkers:s 1 l.
Pickup a passenger going to Equal's Corner.
Go to Equal's Corner:w 1 l.
Switch to plan C if no one is waiting.
Pickup a passenger going to Riverview Bridge.
Go to Sunny Skies Park:n.
Pickup a passenger going to Sunny Skies Park.
Pickup a passenger going to Cyclone.
Go to Riverview Bridge:n 1 r 1 r.
Go to Cyclone:w 2 l.
Pickup a passenger going to Cyclone.
Go to Sunny Skies Park:n 1 r.
Go to Cyclone:n 1 l.
Pickup a passenger going to Multiplication Station.
Pickup a passenger going to Multiplication Station.
Go to Multiplication Station:s 1 l 2 r 4 l.
Pickup a passenger going to Addition Alley.
Go to Rob's Rest:s 1 r 2 l 1 l 1 r 1 r.
Pickup a passenger going to Addition Alley.
Go to Addition Alley:s 1 l 1 l 2 r 1 r 1 r.
Pickup a passenger going to Rob's Rest.
Go to Rob's Rest:n 1 l 1 l 1 l 2 r 1 r.
Go to Cyclone:s 1 l 1 l 2 l.
Pickup a passenger going to Magic Eight.
Go to Sunny Skies Park:n 1 r.
Pickup a passenger going to Cyclone.
Go to Cyclone:n 1 l.
Pickup a passenger going to Cyclone.
Pickup a passenger going to Magic Eight.
Go to Magic Eight:s 1 l 2 r.
Switch to plan D if no one is waiting.
Pickup a passenger going to Addition Alley.
Go to Cyclone:n 1 l 2 r.
Switch to plan B.
[C]
Go to Sunny Skies Park:n.
Pickup a passenger going to Cyclone.
Pickup a passenger going to Addition Alley.
Go to Cyclone:n 1 l.
Switch to plan B.
[D]
Go to Cyclone:n 1 l 2 r.
Pickup a passenger going to Sunny Skies Park.
Pickup a passenger going to Sunny Skies Park.
Go to Sunny Skies Park:n 1 r.
Go to Rob's Rest:s 2 r 1 r.
Pickup a passenger going to Cyclone.
Go to Cyclone:s 1 l 1 l 2 l.
Pickup a passenger going to Cyclone.
Pickup a passenger going to Cyclone.
Go to Zoom Zoom:n.
Go to Cyclone:w.
[E]
Pickup a passenger going to Joyless Park.
Pickup a passenger going to Divide and Conquer.
Go to Joyless Park:n 2 r 2 r 2 l.
Go to Cyclone:w 1 r 2 l 2 l.
Pickup a passenger going to Cyclone.
Pickup a passenger going to Joyless Park.
Go to Joyless Park:n 2 r 2 r 2 l.
Go to Cyclone:w 1 r 2 l 2 l.
Pickup a passenger going to Divide and Conquer.
Pickup a passenger going to Divide and Conquer.
Go to Divide and Conquer:n 2 r 2 r 1 r.
Pickup a passenger going to Magic Eight.
1 is waiting at Starchild Numerology.
Go to Starchild Numerology:e 1 r 3 r 1 l 1 l 2 l.
Pickup a passenger going to Magic Eight.
Go to Magic Eight:w 1 r 2 r 1 r.
Switch to plan F if no one is waiting.
Pickup a passenger going to Riverview Bridge.
Go to Joyless Park:e 2 l 4 r.
Pickup a passenger going to Cyclone.
Pickup a passenger going to The Underground.
Go to The Underground:w 1 l.
Pickup a passenger going to Cyclone.
Go to Fueler Up:s.
Go to Riverview Bridge:n 3 l.
Go to Cyclone:w 2 l.
Switch to plan E.
[F]
Go to Joyless Park:e 2 l 4 r.
Pickup a passenger going to What's The Difference.
Pickup a passenger going to Cyclone.
Go to Cyclone:w 1 r 2 l 2 l.
Pickup a passenger going to Multiplication Station.
Pickup a passenger going to Multiplication Station.
Go to Multiplication Station:s 1 l 2 r 4 l.
Pickup a passenger going to What's The Difference.
Go to What's The Difference:n 2 l 1 r 3 l.
Pickup a passenger going to Knots Landing.
Go to Knots Landing:e 4 r 1 l.
Pickup a passenger going to The Babelfishery.
Go to The Babelfishery:w 1 l.
Pickup a passenger going to Post Office.
Go to Post Office:n 1 l 1 r.

Exits with an error. I could exit properly by going to the Taxi Garage, but that would add 33 more bytes.

Try it online!

I went to a lot of places in Townsburg that I've never visited before for this program. Because this program is HUGE (mainly due to the fact that there isn't a one-stop way to take square roots; I still don't actually do that, though), I'm going to attempt to sketch out an explanation of this program.

Go to Post Office:w 1 l 1 r 1 l.

[Take an input line from STDIN.]

Pickup a passenger going to The Babelfishery.
Go to The Babelfishery:s 1 l 1 r.

[Take it to The Babelfishery, which converts the string to a double.]
[If you were to give it a double, it gives back a string.]

Pickup a passenger going to Cyclone.

[We're going to clone this number by sending it to the Cyclone.]

0 is waiting at Starchild Numerology.

[In Taxi, numeric passengers are introduced once they wait at Starchild Numerology.]
[This passenger will be our running squares total.]

0 is waiting at Starchild Numerology.

[This passenger will be our counter, going from 1 to the inputted number.]
[Yes, it's starting at 0 here, but it is incremented at the start of our loop.]

Go to Starchild Numerology:n 1 l 1 l 1 l 2 l.
Pickup a passenger going to Rob's Rest.
Pickup a passenger going to Addition Alley.
Go to Rob's Rest:w 1 r 2 l 1 r.
Go to Cyclone:s 1 l 1 l 2 l.

[Drop off the squares total at Rob's Rest. We'll pick it back up later when we need it.]
[Also, do the actual cloning of the inputted number.]

[At the start of each loop, our only passenger is the counter, going to Addition Alley.]

[B]
Pickup a passenger going to Divide and Conquer.

[Pick up one of the clones of the inputted number.]
[We will leave the other clone here for later.]

1 is waiting at Starchild Numerology.
Go to Starchild Numerology:n 1 r 3 l.
Pickup a passenger going to Addition Alley.
Go to Addition Alley:w 1 r 3 r 1 r 1 r.
Pickup a passenger going to Cyclone.

[Pick up the number 1, and add it to the counter, then pick up the result.]
[This new passenger is our new counter.]

Go to Cyclone:n 1 l 1 l.
Pickup a passenger going to Sunny Skies Park.
Pickup a passenger going to Divide and Conquer.

[First we pick up the other clone of the inputted number, going to Sunny Skies Park.]
[Sunny Skies Park is basically a First-In-First-Out queue, where we will stick it for later.]
[Then we pick up a clone of the counter which we just dropped off, so we can divide the input by it.]
[We will leave the other clone here for later.]

Go to Divide and Conquer:n 2 r 2 r 1 r.
Pickup a passenger going to Cyclone.

[Divide the inputted number by the counter, and prepare to clone it.]

Go to Sunny Skies Park:e 1 l 1 l 2 l 1 l.

[Drop off our clone of the inputted number at Sunny Skies Park.]

Go to Zoom Zoom:n 1 r.

[Can't forget to get gas!]
[Zoom Zoom is a close gas station, and we have so many credits that one stop here every iteration fills us up completely.]

Go to Cyclone:w.
Pickup a passenger going to Sunny Skies Park.
Pickup a passenger going to Equal's Corner.
Pickup a passenger going to Trunkers.

[First we pick up the other clone of the counter, going to Sunny Skies Park for later.]
[Then we pick up a clone of the division result, going to Equal's Corner.]
[Equal's Corner takes multiple numeric passengers, and returns the value of one of them if they are equal, but otherwise it returns no one.]
[Finally we pick up the other clone of the division result, going to Trunkers.]
[Dropping off a passenger at Trunkers is equivalent to the floor function. We need it here because division results are "exact".]
[(Well, as exact as double precision arithmetic can be.)]
[We will compare the result of this to the exact result of the division.]

Go to Sunny Skies Park:n 1 r.

[Drop off our clone of the counter at Sunny Skies Park.]

Go to Trunkers:s 1 l.
Pickup a passenger going to Equal's Corner.
Go to Equal's Corner:w 1 l.

[Compare the exact division result with the floored division result.]
[If they are equal, then the value of one of them will be waiting.]
[This also means that the counter is a divisor of the input, and we should add the square of the counter to our running squares total.]
[If they are not equal, no one will be waiting, the counter is not a divisor of the input, and nothing should happen.]

Switch to plan C if no one is waiting.

[This is Taxi's only conditional operator.]
[It jumps to a bracketed label (in this case, C) if there is no one waiting at this stop.]
[Of course, there is an unconditional version of this, too.]

[If we are on this path, we are adding the square of the counter to our running squares total.]

Pickup a passenger going to Riverview Bridge.

[We don't actually need the passenger that's waiting here (and leaving it there will not work at all).]
[So what do you do when you have a pesky passenger you don't want? You take them to Riverview Bridge!]
[Riverview Bridge has a lovely view, but passengers dropped off there always seem to fall in the river.]
[This means you don't collect your fare for bringing them there, but at least the pesky passenger is gone.]

Go to Sunny Skies Park:n.
Pickup a passenger going to Sunny Skies Park.
Pickup a passenger going to Cyclone.

[We want to pick up the counter (so we can multiply it by itself). That dang input is in the way, but we still need it.]
[So what do we do? We just take it back to where it was waiting!]
[Yup, even though one would never do this in real life, a passenger's current location is a perfectly legal destination in Taxi.]

Go to Riverview Bridge:n 1 r 1 r.

[So long, Equal's Corner result!]

Go to Cyclone:w 2 l.
Pickup a passenger going to Cyclone.
Go to Sunny Skies Park:n 1 r.

[We clone the counter so we can still use it, then we drop the inputted number back at Sunny Side Park.]
[Again, we will leave the other clone here for later.]

Go to Cyclone:n 1 l.
Pickup a passenger going to Multiplication Station.
Pickup a passenger going to Multiplication Station.
Go to Multiplication Station:s 1 l 2 r 4 l.

[Square the counter.]

Pickup a passenger going to Addition Alley.
Go to Rob's Rest:s 1 r 2 l 1 l 1 r 1 r.
Pickup a passenger going to Addition Alley.
Go to Addition Alley:s 1 l 1 l 2 r 1 r 1 r.
Pickup a passenger going to Rob's Rest.
Go to Rob's Rest:n 1 l 1 l 1 l 2 r 1 r.

[Pick up the running squares total from Rob's Rest, add it to the square of the counter, then bring it back.]

Go to Cyclone:s 1 l 1 l 2 l.
Pickup a passenger going to Magic Eight.

[Pick up the counter, going to Magic Eight.]
[Magic Eight takes two numeric passengers, and returns the first passenger if it is less than the second, but otherwise it returns no one.]
[We will compare the counter to the inputted number...]

Go to Sunny Skies Park:n 1 r.
Pickup a passenger going to Cyclone.
Go to Cyclone:n 1 l.
Pickup a passenger going to Cyclone.
Pickup a passenger going to Magic Eight.

[...who we take now, so we can make a clone of it and compare it to the counter.]

Go to Magic Eight:s 1 l 2 r.
Switch to plan D if no one is waiting.

[Compare.]
[If the counter is waiting here, the loop is to continue.]
[If not, switch to plan D, end the loop, and start detecting whether or not the running squares total is a square itself.]

Pickup a passenger going to Addition Alley.
Go to Cyclone:n 1 l 2 r.
Switch to plan B.

[Reset everything, and loop back to the start of plan B.]

[C]

[If we are on this path, the counter is not a divisor of the inputted number, but the loop should continue anyways.]

Go to Sunny Skies Park:n.
Pickup a passenger going to Cyclone.
Pickup a passenger going to Addition Alley.
Go to Cyclone:n 1 l.
Switch to plan B.

[Reset everything, and loop back to the start of plan B.]

[D]

[We are finally out of the loop. Now it's time for the real fun.]

Go to Cyclone:n 1 l 2 r.
Pickup a passenger going to Sunny Skies Park.
Pickup a passenger going to Sunny Skies Park.
Go to Sunny Skies Park:n 1 r.

[We don't need the inputted number anymore, but we can't change its destination from the Cyclone.]
[So we go to the Cyclone, so we can take the two clones and bring them to Sunny Skies Park forever.]
[We could send them to Riverview Bridge, but that will take more bytes.]

Go to Rob's Rest:s 2 r 1 r.

[Pick up our running squares total from Rob's Rest, so it can go out and see the town.]

Pickup a passenger going to Cyclone.
Go to Cyclone:s 1 l 1 l 2 l.
Pickup a passenger going to Cyclone.
Pickup a passenger going to Cyclone.
Go to Zoom Zoom:n.
Go to Cyclone:w.

[We need to end up with 4 clones of the total.]
[Two of them will actually be used as the counter in this loop.]

[At the start of each loop, we have no passengers.]

[E]
Pickup a passenger going to Joyless Park.
Pickup a passenger going to Divide and Conquer.
Go to Joyless Park:n 2 r 2 r 2 l.

[We will leave one clone of the total at Joyless Park (like Sunny Side Park, but on the opposite side of town).]
[The other clone will be used in a division.]

Go to Cyclone:w 1 r 2 l 2 l.
Pickup a passenger going to Cyclone.
Pickup a passenger going to Joyless Park.
Go to Joyless Park:n 2 r 2 r 2 l.

[We will leave one clone of the counter at Joyless Park.]
[We will clone the other clone (clone-ception?).]

Go to Cyclone:w 1 r 2 l 2 l.
Pickup a passenger going to Divide and Conquer.
Pickup a passenger going to Divide and Conquer.
Go to Divide and Conquer:n 2 r 2 r 1 r.

[We will use both clones of the counter in a division.]
[Divide and Conquer can take 3 arguments, and the result here will be equivalent to total / counter^2 .]
[If the square of the counter is equal to the total, the result will be 1.]
[If the square of the counter is less than the total, the result will be greater than 1.]
[If the square of the counter is greater than the total, the result will be less than 1.]

Pickup a passenger going to Magic Eight.
1 is waiting at Starchild Numerology.
Go to Starchild Numerology:e 1 r 3 r 1 l 1 l 2 l.
Pickup a passenger going to Magic Eight.
Go to Magic Eight:w 1 r 2 r 1 r.
Switch to plan F if no one is waiting.

[We compare the result of the division against 1 to decide our next course of action.]
[If it is greater or equal, the loop should end, and the counter is equal to the floor of the square root.]
[If it is less, the loop should continue.]

Pickup a passenger going to Riverview Bridge.

[Whoops, we don't actually need this anymore.]

Go to Joyless Park:e 2 l 4 r.
Pickup a passenger going to Cyclone.

[We will clone the total later, so we can reuse it in the next loop iteration.]

Pickup a passenger going to The Underground.
Go to The Underground:w 1 l.

[Instead of taking the number 1 and subtracting it from the counter, we can do an optimization here.]
[The Underground is a destination that takes 1 passenger, and subtracts 1 from it.]
[If the result is positive, it returns the result. Otherwise, it returns no one.]
[The latter result will not happen for us, however.]

Pickup a passenger going to Cyclone.

[We will clone the counter later, so we can reuse it in the next loop iteration.]

Go to Fueler Up:s.

[Gotta remember to stop for gas!]

Go to Riverview Bridge:n 3 l.

[So long, Magic Eight result!]

Go to Cyclone:w 2 l.
Switch to plan E.

[Reset everything, and loop back to the start of plan E.]

[F]

Go to Joyless Park:e 2 l 4 r.
Pickup a passenger going to What's The Difference.
Pickup a passenger going to Cyclone.

[Once we've figured out the square root of the total, we're still not done.]

Go to Cyclone:w 1 r 2 l 2 l.
Pickup a passenger going to Multiplication Station.
Pickup a passenger going to Multiplication Station.
Go to Multiplication Station:s 1 l 2 r 4 l.

[We actually square the square root...]

Pickup a passenger going to What's The Difference.
Go to What's The Difference:n 2 l 1 r 3 l.

[...and subtract it from the total at What's The Difference (which subtracts dropped-off passengers).]

Pickup a passenger going to Knots Landing.
Go to Knots Landing:e 4 r 1 l.

[If the result is 0, the total is a perfect square, and a truthy value should be output.]
[If the result is nonzero, the total is not a perfect square, and a falsy value should be output.]
[Luckily, Knots Landing does our work for us.]
[Knots Landing inverts the boolean logic of numeric passengers (i.e. 0 becomes 1, nonzero becomes 0).]

Pickup a passenger going to The Babelfishery.
Go to The Babelfishery:w 1 l.

[Convert it to a string...]

Pickup a passenger going to Post Office.
Go to Post Office:n 1 l 1 r.

[...and output it.]

[We would exit the program by going to the Taxi Garage, but that takes bytes.]
[The boss fires us because we didn't bring the taxi back to the garage, but that gets output to STDERR, so that's OK.]