Egg binding occurs when a hen who is forming an egg in her uterus does not have enough calcium to finish forming the egg shell. The shell does not become hard enough to push out, which causes the hen extreme stress and extreme risk.
To prevent egg binding it is very important to have a cuttlebone and mineral block available in the cage, even if they are not breeding (sometimes a hen will spontaneously start laying eggs without a nest box or even without a mate). Especially when a pair is breeding, it is important to regularly check if the cuttlebone and mineral block have been used up. If so, they need to be replaced immediately. If a breeding pair does not touch the cuttlebone or mineral block, a calcium supplement may be necessary.
Article on egg binding
When a hen becomes egg bound, I have had success with holding the rear end of the hen in a bowl of fairly warm water. It has take sometimes up to 20 to 30 minutes to pass the egg, but I haven’t lost one yet. If you try to manipulate the egg and it breaks, then the hen will die.
Another idea I have heard is to use Preparation H ointment on and around the cloaca with a fair amount of success. If the hen still can’t lay the egg, you will have to find medical care for this bird (an avian vet) or you will lose the hen.