I think the answer to your question is 'no', but as I will point out below it is possible that there is some clever way to make proton production happen.
The first thing to note is that photons are always easy to produce as "soft" excitations due to their masslessness. Massive excitations are significantly more expensive to produce.
This doesn't yet address your specific question about protons. In the usual formulation of the Casimir effect, it is discussed within quantum electrodynamics (QED). The fundamental excitations of QED are the photon and electron/positron. In this context the dynamical Casimir effect would be expected to produce photons and electron/positron pairs, but there is no mechanism for generating other particles. And, of course, photon generation will have qualitative differences from the pair production processes due to the difference in particle masses.
If you want to think about protons, it is better to think about the dynamical Casimir effect in the context of quantum chromodynamics (QCD), which is a context where I have not personally seen it considered. Presumably, virtual particle production proceeds there as in QED, but again, the proton is not a fundamental particle of the theory. The new particles which could be produced directly are quarks and gluons. Protons are bound states of such particles, so proton production would require a subsequent binding of the particles produced by the dynamical Casimir effect. Perhaps there is a a clever way to arrange for this to happen, but it seems to me that protons cannot be produced directly.