Just an update on the hen
She seems to be perfectly okay now. She did not have worms, and after just sitting and watching her it appeared that her vent muscles just could not push out the poop properly. She laid an egg two days ago, and I am assuming it is the first time to lay an egg in a month or two. She ripped her vent a bit but now her muscles seem to be perfectly functional. I still wipe her off daily because she bleeds from the rips whenever she lays an egg, but she no longer has a poopy butt. The rips seem to be healing with no obvious sign of infection.
I cleaned her today and while holding her she pooped and everything seemed functional with solid, normal chicken poop.
So I don't really know what her overall problem was, but she seems to be better.