How do I cut a square in the middle of a Plexiglas sheet?

You'll need a hand saw with a removable blade (like a hacksaw) and a drill bit that is a slightly larger diameter as the saw blade. Sketch the square in the location you want it on the material, then drill into the four corners of the square, taking care to make sure the hole doesn't cross over the square boundary you drew. Next, fit the blade through the hole and proceed to cut along the lines you've drawn. Cut the square a bit smaller than your final dimensions so you can go back with some sandpaper to clean up the cut, and use a sharp blade (use gloves!) to deburr and/or cut a chamfer into the edges.