I had a small water leak in tbe bathrom that wet the plywood a little and made it swell a little. I am going to put backerboard down over the old floor and am wondering if peel and stick vinyl squares will stick to backerboard? Also, will it be OK to put the backerboard over the old plywood flooring?

Yes but I usually glue and screw the new backer board. I'd suggest Hardiebacker or Wonderboard both 1/4'. Peel and press tiles will stick better if you apply 2-3 coats of adhesive primer to the backer, same aisle in the store as the tile. I glue with acrylic modified thinset and screw with 1.25" special screws for Hardiebacker.

Good call ballenger!
Ben you do need to do something where the boards will butt to each other. The vinyl tile will eventually shadow through what is under it. The butt joints need to be filled so the floor is even and as long as you are patching the joints might as well go over the sunken screw heads won't take much for a high heel or similar to punch a hole if it is placed just right. Also be sure the plywood that got wet is dry this board plus the primer that ballenger recommends will not allow moisture to go through it. Granted it won't hurt this board byt it will hurt the joists it is on and if this is a second floor the drywall on the ceiling below it.