If there was digging, very unlikely to be a cat. What kind of coop is at the other house? Could you just move it to yours? I know you said you have a broody hen, but it might be possible, if that is a secure coop, that it would be better and cheaper and faster to simply move it. Simply, she says, lol.
Hens go broody when you don’t want them to… and won’t go broody when you do. 