DUCT TAPE!! Duct tape fixes everything!!
Just kidding but we had a similar problem with Sammy for awhile and I finally put duct tape around the seam where the top and bottom hook together. I didn't put it on really tight because you have to remove it to open the box, but it kept everything inside where it belonged. I would say that any type of wide tape would work for this situation because you really don't need it to hold the boxes together. I found it easier than putting a bunch of stuff under the box. Of course all this was prior to the invasion of many more kitties. Now we have huge boxes with no lids. Hope this helps and good luck!!