TI-Basic, 8080 83 bytes

Input N
For(I,0,N
For(J,0,N
For(K,0,N
For(L,1,N
If I²+J²+K²+L²=N and not(Ans(1
{L,K,J,I}²→S
End
End
End
End
sum(not(not(Ans→dim(ʟS
ʟS

Output is stored in ʟSAns and displayed.

LongerFaster but fasterlonger solution:

100100 103 bytes

Input N
√(N
For(I,0,Ans
For(J,0,Ans
For(K,0,Ans
For(L,1,Ans
If I²+J²+K²+L²=N:Then
{L,K,J,I}²→S
√(N→I
Ans→J
Ans→K
Ans→L
End
End
End
End
End
sum(not(not(ʟS→dim(ʟS
ʟS