Not necessarily... In the example you give - a ball in three dimensions with Euclidean metric - there are continuous families of periodic orbits of all numbers of reflections $n\geq 2$ (polygons and stars on planes passing through the origin). But a typical perturbation of the metric would break the symmetry and leave only a finite number of periodic orbits for each $n$. So most of the original periodic orbits are unstable in this sense.
Edit: In response to the question edit: Persistence of a periodic orbit is a local question, depending only on the metric and boundary properties on/near the orbit. It cannot "see" the global symmetry if any. So, the unperturbed manifold need not be symmetric away from the periodic orbit. Furthermore, an isolated non-hyperbolic periodic orbit may also vanish under perturbation of Hamiltonian systems, in for example a saddle-center bifurcation. Such isolated non-hyperbolic orbits may be obtained easily in convex billiards by tuning the local boundary curvatures.