I posted a similar problem last week, got only one response.
The pic of your hens butt looks just like my hen. I followed other threads info and put her in a warm bath, and prep H on a gloved finger and removed what looked like the yolk of a hard boiled egg about the size of a superball , then kept her warm and quiet and she passed 2 more smaller "soft" eggs. Now she is with the others in the coop, she is moving slowly, but eating and drinking. Still is acting like there is something stuck in her butt, and find her in the nesting box occasionally when I check on her. I guess I'm just taking a wait and see attitude at this point.