Proof for Product of Congruences mod P

Question

Consider $Z_p$, with $p$ prime. Prove that $[x][x] = [y][y]$ if and only if $[x] =[y]$ or $[x] =[-y]$.

I think this question comes down to the fact that $Z_p$ is cyclic but am not sure.

Answer 1

Hint: One direction is easy, and does not require primality.

From a number-theoretic point of view, the harder direction boils down to the fact that $p$ divides $x^2-y^2$ if and only if it divides $(x-y)(x+y)$.

And if a prime divides a product, it divides (at least) one of the terms.