When we have y² = z² (mod n) then y² − z² = kn for some k.
But y² − z² = (y+z)(y−z) = kn. We have ruled out the case where n divides y+z or y−z as trivial.
(y+z)(y−z) = kn and thus n divides (y+z)(y−z), but divides neither (y+z) nor (y−z), therefore some factor of n must divide (y+z) while its cofactor divides (y−z).
Thus gcd(n, y+z) and gcd(n, y−z) must both yield factors of n greater than 1.