Terry Rockafellar has a paper on this topic in SIAM Optimization titled Generalized Hessian Properties of Regularized Nonsmooth Functions. See theorem 3.8. Rockafellar proves an equivalence between twice-differentiability of the Moreau envelope and twice epi-differentiability of the function $f$.

https://doi.org/10.1137/S1052623494279316

And for posterity's sake, $z$ should be $h$ in your definition of the envelope. This has been noted already in the comments.

Terry Rockafellar has a paper on this topic in SIAM Optimization titled Generalized Hessian Properties of Regularized Nonsmooth Functions. See theorem 3.8. Rockafellar proves an equivalence between twice-differentiability of the Moreau envelope and twice epi-differentiability of the function $f$.