The one thing you should always do first when egg binding is suspected is to give a calcium tablet whole directly into the hen's beak. It encourages contractions to help your hen get the egg out. Lubing the inside of the cloaca can also help make things slippery so the egg slides out more easily.
Coincidentally, I am presently treating one of my own hens for this. This hen is getting up in years, around seven now, and she isn't laying normal eggs. They are all soft and shell-less. These are very hard to pass. She's been on a daily calcium citrate tablet for a couple weeks trying to improve her shells. She got a calcium this morning, but this evening, I saw she was in crisis mode.
I popped another calcium into her beak and brought her inside and placed her in a crate on a bunch of towels warmed in the dryer. Inside of thirty minutes she passed a collapsed membrane. She's still probably got egg material inside her so she's still in the crate in the garage tonight. By morning, she should have passed the rest of the egg or even a second egg, which is more common than you might think.
I will go by her behavior. If she's acting normal, that means she's expelled it all. If not, she will get more calcium and an oral antibiotic.
So give your hen one of these right now and give her something warm to rest upon. Hopefully the egg will come out okay.
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