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##Python, 133 129 bytes

from itertools import*
n=input()
l=[];s=0
for i in range(-n,n+1):l.append(i**2)
for j in product(l,l):
 if sum(j)==n:s+=1
print s

This demonstrates the use of itertools cartesian product. Try it online

Python, 133 129 bytes

from itertools import*
n=input()
l=[];s=0
for i in range(-n,n+1):l.append(i**2)
for j in product(l,l):
 if sum(j)==n:s+=1
print s

This demonstrates the use of itertools cartesian product. Try it online

##Python, 133 bytes

from itertools import*
n=input()
l=[];s=0
for i in range(-n,n+1):
    l.append(i**2)
for j in product(l,l):
    if sum(j)==n: s+=1
print s

This demonstrates the use of itertools cartesian product. Try it online

##Python, 133 129 bytes

from itertools import*
n=input()
l=[];s=0
for i in range(-n,n+1):l.append(i**2)
for j in product(l,l):
 if sum(j)==n:s+=1
print s

This demonstrates the use of itertools cartesian product. Try it online

##Python, 133 bytes

from itertools import*
n=input()
l=[];s=0
for i in range(-n,n+1):
    l.append(i**2)
for j in product(l,l):
    if sum(j)==n: s+=1
print s

This demonstrates the use of itertools cartesian product. Please help with the golfing! Try it online

##Python, 133 bytes

from itertools import*
n=input()
l=[];s=0
for i in range(-n,n+1):
    l.append(i**2)
for j in product(l,l):
    if sum(j)==n: s+=1
print s

This demonstrates the use of itertools cartesian product. Try it online