Density questions on primitive divisors of Lucas sequences

It is known that every term $U_n$ of a regular Lucas sequence has a primitive prime divisor if $n\ge 31$, i.e., a prime $p$ such that $p\mid U_n$ but $p\nmid U_k$, for all $1\leq k<n$. Can this be refined to specific sets of primes?
We ask a weaker question: what proportion of terms $U_n$ possesses a prime factor $p$ such that $Q$ is a quadratic non-residue modulo $p$? Here $Q$ is the second parameter of the Lucas sequence. We find a natural density of either $1/2$ or $1/4$ in many cases. As a consequence, infinitely many terms $U_n$ have a primitive prime divisor $p$ with $(Q/p)=-1$, with a positive lower density for certain values of $Q$. However, many cases remain in which our methods fail to apply. Exploring other ideas leads to other interesting density questions.