My hens also get pasty butts in the rainy season here in the Philippines, but when I clean their butts up it doesn't get dirty anymore. Yes, Sunshine might have Diarreha(stop giving her veggie treats for a while and feed oatmeal).
As to bietsch624, the same thing happened to me. My little hen Rexie was happy and laying about, but suddenly she seems sick and has a pasty butt. We washed her butt, and the next day she looked ok again. Well, 7 months later, she hasn't laid ever since the day we washed her butt. She got sick again, but now I noticed red blood patches on her foot, and again a pasty butt. I washed her butt clean, but the next day she died. We cut her body open and we found a very old, rotting egg that was stuck in her for 7 months. The bacteria from the rotting egg must have made her very sick.
It might be caused by egg binding, where an egg gets stuck in the hen's vent and has a hard time laying it. The stuck egg might have clogged the poo and caused her to have a pasty butt. Just a theory, but I'm very sorry for your loss. I, too, was crushed when Rexie departed.
To cure egg binding, gently massage the hen's but in a way like your forcing something to come out. It should come out soon, but if this doesn't work, if the egg is visible in the vent, gently poke and break it with a pencil and very carefully remove the egg shell bits to prevent injury in her innards. Apply some bandage if some flesh stick out of her vent as the other chickens might peck on it.
Good luck!