Build your 2x2 box with a rebate around the inner edge the same depth as the plexi. Drill holes in the rebate near each each corner to accept a rubber grommet. The grommets sit slightly proud of the rebate by a small fraction of an inch. Then place the plexi into the 'frame' and use a very slight flex in the plexi itself to trigger thin contact strips (or micro switches) on at least two inside edges . Use at least two switches in parallel to ensure triggering regardless of foot placement.
Go a step further, Wire two or more Led "looms" to separate circuits i.e. one per switch and get different colors lighting up depending on where you stand!

Use a diffusing material (thick PVC) below the plexi to get a smooth, even diffusion of light.
This of course assumes the plexi is reasonably springy and will tolerate repeated (but a very small) amount of flex without cracking.