Cardboard is snubbed by hens. My DH tried to set up a big cardboard box in the yard and they didn't come near it. However, a large plastic covered doghouse is frequented by our hens. They once used it to dive into when they sounded a Cooper's Hawk alarm. Since then I've added an even larger recycled doghouse for them. During heatwaves they don't often stay long inside the doghouses but during cooler temps they sometimes take mid-day snoozes in them. I keep straw in them to make cleaning out poops easier.
For nests inside the indoor 4x4 pen we placed a couple Kidz Homz (Walmart online) storage boxes that have lids and one open side wall and the hens love them to sleep in or lay their eggs as long as there's straw in them. Hens love seclusion so open nestboxes or plastic tubs aren't appealing. However a covered kitty litter pan with a round hole entrance is very appealing to them. Your plastic kitty litter bucket would probably have worked if it had a lid and a round hole entrance. Some owners who had very open nestboxes gave their hens more seclusion by putting up curtains over each nestbox. Sounds silly but it worked for them. I've seen all sorts of patterns and styles of curtains from little ruffles on the top front to fully parted drapes on each side of each nestbox. The idea is to offer seclusion.
I have two from them now, and one's slightly darker than the other, but both are super sweet! 
