I don't know. Is there any chance you can put a roost in there to get them off the floor, rather than the box? I can relate to the wet. We've been raining for days. I HAVE to wear my boots out there whenever I go out.
I am sorry I can't tell you what the white spots are. Hopefully someone will know. Good luck and I'll keep searching to help you.
Do you have any pictures? It may be Thrush or a type of yeast infection. I have been trying to figrue out what is wrong with my Hens rear and a lady sent me a link to this site. See if it fits what you have. I don't think it's what mine has.