the poo and egg come out of the same hole. but from different places inside her body, I would bring her inside, If you decide to soak her the reason being they say it helps them to relax and if theres an egg in there it might just come out. but don't put her back outside wet, it would be best just to make her a quiet place in the house. I am going to be honest it doesn't sound good, the egg wouldn't be red or fleshy but if she's straining to push an egg out your just seeing the inside of her vent. sometimes when a hen lays an overly large egg they can get whats called prolapse that means part of the inside can come outside. thats not what you have but what your seeing is the part that would come out most likely, so don't do anything else as far as trying to feel up inside her vent. I have never had to deal with an egg bound hen so I am giving to advise from what I have read on here..... this is very much an emergency