This is a common problem in domestic poultry. To confirm that this is egg binding, gently feel inside the cloaca for an egg. If one is readily felt, it is likely the source of the problem.
The problem is usually caused by an imbalance of Calcium/Phosphorus/Vitamin D in the diet. This can be because of free ranging or the feeding of a diet high in seeds and other treats. In high producing breeds, sometimes a well balanced diet just wont provide enough calcium, and oyster shells should be available at all times.
These birds are usually cold and dehydrated, so the first step is to bring her in and warm her up. Keep water in front of her, but don't force her to drink. A vet would be able to give her injectable calcium to help her pass the egg, but if that is not an option here's what you can do:
Get some KY Jelly (water soluble lube) from the pharmacy. Put a quarter sized dollop into the cloaca using a very clean finger (or use latex gloves). Close the cloaca and massage the KY trying to get it up and around the egg. Try to massage the egg out of the bird using your hands on the abdomen. Be gentle, as this may take a while. a finger in the cloaca can help the egg to move along.
If you cannot get the egg to pass, you can get a needle and syringe and drain the egg, then gently collapse the shell. You risk lacerating the cloaca with this procedure, but it is a small risk. This is necessary if you cannot get the egg to pass.
Care of the bird afterward includes keeping her inside and warm with food and water available. Also make sure a calcium source is available. Then keep observing her until she passes a normal egg.
Try to correct any underlying problems to prevent the issue from reoccurring.
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