For small values of $n$ and $(a, n) = 1$ it is sometimes possible to give an elementary proof that there are infinitely many primes congruent to $a \bmod n$ along the lines of Euclid's classic proof of the infinitude of the primes. The idea is to find a non-constant integer polynomial $p(t)$ such that the primes dividing an integer value of $p$ are, with finitely many exceptions, congruent to $a \bmod n$ or $1 \bmod n$, and such that the first case occurs infinitely many times; then one proves, just as Euclid did, that the integer values of $p$ are divisible by infinitely many primes, and then one contrives to avoid the residue $1 \bmod n$ (if $a \neq 1$; if $a = 1$ then we take $p(t) = \Phi_n(t)$.) Results of Schur and Murty (see this paper of Keith Conrad) then imply that such a polynomial $p$ exists if and only if $a^2 \equiv 1 \bmod n$.

However, Euclidean proofs are capable of identifying certain extra subsets $S \subset (\mathbb{Z}/n\mathbb{Z})^{\ast}$ such that there are infinitely many primes congruent to an element of $S \bmod n$. For example, I think we can take $S$ to be the complement of any proper subgroup of $(\mathbb{Z}/n\mathbb{Z})^{\ast}$ (in particular when $n$ is prime we can take $S$ to be the set of quadratic non-residues).

Question: Is there a characterization of the subsets $S$ for which this is possible, generalizing the results of Schur and Murty? In particular if $S$ and $S'$ are two such subsets with non-empty intersection, is $S \cap S'$ such a subset?

(If this question is somehow answered in Conrad's paper, my apologies; the material is somewhat beyond me.)