# [Pyth], 15 bytes

    }Qmjk^R2d*SsQUs

[Try it online!][TIO-lzvbr7wq]

[Pyth]: https://github.com/isaacg1/pyth
[TIO-lzvbr7wq]: https://tio.run/##K6gsyfj/vzYwNys7LsgoRSu4ODC0@P9/JUsDJQA "Pyth – Try It Online"

[Test cases][TIO-lzu9a98v]

[Pyth]: https://github.com/isaacg1/pyth
[TIO-lzu9a98v]: https://tio.run/##K6gsyfj/36c2KTcrOy7IKEUruDgptDjpf6VvtJKhoZKSpYGSkqGBAZBlaGYCYgJZBoZmQAkgMjRR@g8A "Pyth – Try It Online"

A program that takes an integer value as a string (because the trailing 0 is recognized as a a separate value) and outputs `True` for the 'concats of two squares', `False` otherwise. The calculation time for the large values exceeds 1 minute.

#### Explanation:
```python
}Q              # check if the input value is in the list.
  m             # The list is constructed by mapping:    
   jk           # concatenation
     ^R2        # and squaring
        d       # of the argument d
         *      # mapped over the cartesian product
          SsQ   # of the range from 1 to Q
             Us # with the range from 0 to Q-1
```