Problem is, IF you could create a perfect mirror in the shape of a sphere
(It would have to be a sphere not a square i think right?)
If this thing can open, it is no longer perfect
So even if you can close it really really fast, it still is not a perfect mirror
Also, doesn’t it have to be an absolutely perfect vacuum inside?
No matter at all in it except the photon?
Yes, and this is a problem (well, another problem) with the idea because the concept of a perfect vacuum is a classical simplification, and not one which actually exists.
By the description of the experiment, you have one or more photons inside the box, or sphere, or other container. That means, at the quantum level, you have an excited photon field. That photon field is coupled to the electron and positron fields, so there’s a certain probability that you’ll spontaneously get an electron and a positron as virtual particles, as some of the energy from the photon field gets transferred into those other fields and back. Those virtual particles can interfere with the photon field in other ways, because nothing says they can’t, so over enough time they will.
More to the point, your perfect mirrors have to be made of something. Something made of baryons and leptons, if it’s matter as we know it. Something made of particles coupled to the photon field. Something made of particles which cannot be at absolute zero, both because they’re on the edges of a region which just had light shone into it and because there is no way to cool matter to absolute zero even in principle. That destroys the whole idea of a perfect mirror, really: Once you admit what we know about quantum physics into this thought experiment, it all falls apart.