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The only thing holding the door square and plumb is the drywall. You
should have a diagonal from the top of the hinge side to the opposite
loiwer corner. The other question is warping across the surface. Even
a light hollow core door will have a lot wider rails and stles than
your 2x4s on edge. Gluing the drywall in will make a stiffer box

If I'm following what you're saying, either you misunderstood my description
or I didn't describe it well enough.

The 2x4's are all going to be joined along their widest part oriented front
to back- making the depth (thickness front to back) of the frame without
the drywall 3 1/2", 4 1/2" after adding drywall on both sides, which is the
same thickness as the rest of the booth - i.e., this will be like a section
of the front wall of the booth, except it will be on hinges.

To further elaborate, I'll have two 12-3" screws at each point where the
frame members join. A 44 1/2" piece across the top and bottom, with three
78" pieces between them vertically - one on each side and one in the center.
The drywall will be screwed into the edges of all of these. For added
rigidity and support I figured I'd add at least one diagonal member with the
ends cut at 45 deg. to put on the inside of the frame at the upper right
corner on the same side as the hinges.

In reality, I'll probably shorten all these dimensions by about 1/8" just to
allow a small gap all around, with stops mounted on the inside frame of the
box. I plan to hold the door closed with magnets mounted flush at strategic
points on either the outer edge of the door or the stops along with a strip
of metal mounted on the opposite side from the magnets.

I'll be picking the straightest pieces of lumber on and around the door area
and shaving off just a tad with a table saw to make it as close to a
perfectly mated area as possible. All the cuts on the 2x4's have been made
with a miter saw and are all accurate to probably within half the width of
mechanical pencil lead diameter of each other.