Does generalization of axioms apply also to theorems? In Enderton's book "A Mathematical Introduction to Logic" (second edition), he includes six axiom groups, and allows also for a generalization of those axioms such that if $\Psi$ is an axiom then $\forall x \Psi$ is also an axiom.
Is this rule also intended to apply to theorems from those axioms?  In other words, if $\Psi$ is a theorem of $\Gamma$, can I conclude $\forall x\Psi$ is also a theorem?

If the logic in question is halfway sane (such as if its entailment relation agrees with ordinary first-order logic) then yes -- except, possibly, if $\Gamma$ contains $x$ free. You don't always get to infer $\forall x.p(x)$from $p(y)$.
Proving that this is the case depends on details of the proof system that you have not given, though. In some systems it is a primitive rule that $\psi$ entails $\forall x.\psi$.
