Unless you can place the box away from where they normally urinate and defecate in the cage, they will foul this box and bedding with urine and feces. Unless you were to clean this out daily, the wet bedding would create more discomfort than sitting on a board in a cage. Having a rabbit wet and cold is far worse than merely cold. Now, if you were to have a box with a wire bottom and a sitting pad, this may block the wind somewhat and still be sanitary.
Not to split "hares", so to speak, but keeping ANYTHING in a cage is cruel. So, if one must keep an animal in a cage, trying to compensate for this by providing lux digs is kind of silly.
Before you get all up in arms thinking I am criticizing keeping rabbits in cages...I'm not. I used to keep meat rabbits in cages suspended in a small rabbit barn. They have fur coats, folks. They actually thrive in the winter more so than in the summer. Mine did very well with their cages butted up against each other, plenty of hay in their hay nets for nibbling, water from nipple waterers, and clean cages.
I only used boxes for rabbits about to have young. They will line this box with so much fluff that the babies are living in a hot box!