Does Poincare recurrence imply that a photon shot into a box will exit the way it came in? If you have a big closed cube that has perfectly mirrored surfaces on the smooth flat walls or faces of the cube and only one corner has a tiny 'entrance', a narrow hole at a specific angle, say 'pointed' at a spot a few millimeters from the cube's center; you 'fire' a photon through the hole and it 'bounces' around for a long time. Eventually will it 'get back to the hole and at the correct angle emerge from the cube?
 A: The answer is maybe. Heavily paraphrased, Poincaré recurrence says that if a bounded system starts in a certain state, then it will at some point in time return to a state arbitrarily close to that original state. It does not say anything in particular about other states that the system may take (unless you know that it will have some other state in the future, in which case you could call that the "starting" state instead). Also, instead of "photon", let's just say its a particle moving at the speed of light. Real photons will have nasty interference physics when trying to pass through the small hole in the box.
Case 1: The particle starts in one corner of a closed box with an initial velocity. It "leaves" when it has the same position with the opposite velocity.
Both the positions and velocities are bounded, so Poincaré recurrence applies to this system. However, it only predicts that the particle will return to that original state (as well as any state that we know it ends up in). Whether or not it can arrive at a state with the same position and opposite velocity depends on exactly what trajectory the particle takes in the box, which in turn depends on exactly what its initial state was. For instance, if the particle is initially pointed towards the exact opposite corner of the cube, we can expect it to bounce back to the starting corner with the opposite velocity. Note, though, that this is a question of geometry, not of Poincaré recurrence.
Case 2: A particle starts outside a closed box with an infinitesimally small hole in the corner. It has an initial velocity vector that lets it pass through the hole. Outside the box is an infinite universe.
In this case, the velocities are still bounded, but the positions are not. In principal, the particle could take any position at all. Poincaré recurrence does not apply. Whether or not the particle leaves the box is again determined by geometry and its initial state.
Case 3: Same as Case 2, but how the cube-plus-particle system sits inside another enclosure, maybe a laboratory.
Now the particle positions are bounded, so Poincaré recurrence at least applies. Furthermore, since the particle starts outside the cube, Poincaré recurrence says that it must somehow, someday end up in that same position with the same velocity. So in this case yes, recurrence predicts that the particle will exit the box.
