Yikes!
I am sure people with more cat experience can help you and my idea might be vetoed, but in a pinch what about those table clothes you can buy ( think I got one a walmart for $1.50) with the plastic on top and like felt underneath? laying them underneath the boxes? That might work in a pinch but I am leaning towards the problem is behaivoral and I know that I can't help you there.
good luck.
Staci