Here's a sketch of a solution:
Assume that the set of primes satisfies the conditions of the problem. It suffices to find a prime such that is a quadratic residue mod for . That's because the law of quadratic reciprocity implies that each is then a quadratic residue mod .
Consider the set of congruences
Any solution is a quadratic residue mod for all . By the Chinese Remainder Theorem, a solution exists and all solutions are of the form , where and . By Dirichlet's theorem, this arithmetic progression contains a prime , as required.