This is an old question (and according to this MO question, the result you seek was proven by Landau). In particular, it follows from this that if $S$ is a set of arithmetic progressions containing a density $1/2$ set of the primes, then number of $n \leq x$ all of whose prime factors in $S$ is asymptotic to some constant times $C \frac{x}{\sqrt{\log(x)}}$. This implies that the $n$th positive integer $q_{n}$ all of whose prime factors are $\pm 1 \pmod{8}$ satisfies $q_{n} \sim C_{1} n \sqrt{\log(n)}$. Similarly, if $r_{n}$ is the $n$th positive integer all of whose prime factors are $\pm 3 \pmod{8}$, then $r_{n} \sim C_{2} n \sqrt{\log(n)}$.

If you are looking for a modern treatment of this, which will enable you to compute the constants $C_{1}$ and $C_{2}$, this result is stated in Cojocaru and Murty's book "An introduction to sieve methods and their applications", but I don't have my copy at home right now, so I can't tell you exactly where in the book to look.