One does not have to dig too deep into the arguments to answer this. Maynard's formulation (see his preprint [here][1]) shows that if $x$ is sufficiently large, there is a Zhang prime (or even a prime $p$ so that $p+k$ is prime with $k \leq 600$) between $x$ and $2x$. It follows that there is a constant $C$ so that $z_{n} \leq C \cdot 2^{n}$, which is primitive recursive. Often in analytic number theory, it's difficult to show that something happens without showing that it happens "a lot".


  [1]: http://front.math.ucdavis.edu/1311.4600