My question has been here on MSE for a long time, but it has not received a full answer. I bring it here:

Find a prime $p$ such that $p \equiv 1 \bmod 4$ and such that the product in the range $[1,p-1]$ of the quadratic residues + the product in the range $[1,p-1]$ of the quadratic non-residues is divisible by $p^2$, or show that such a prime does not exist.

Nota: I know that for any prime $p$ such that $p \equiv 3 \bmod 4$, the product of the quadratic residues + the product of the quadratic non-residues is always divisible by $p^2$. I have check that this is not the case for all primes $p$ smaller than $400000$ and such that $p \equiv 1 \bmod 4$.