There are a few common reasons a hen can become egg bound - one is that the egg is just too big for the pelvis, another possibility is there is not enough calcium in the diet (calcium is required for muscle retraction), another possibility is that she might be too weak from some other condition (or too fat) which could be nutrition or illness related, and sometimes she may be dehydrated and the part which moves around the egg as it is laid is too dry and stuck to it preventing it from getting out.
Signs that you may have a chicken that is egg bound could be repeated visits to the nest and/or trying to lay an egg for hours and becoming distressed. Later on she may become lethargic and 'droopy'. If you have noticed your hen acting strangely in this way (you may even get alerted to it by one of her coop mates who has raised the alarm) you can check for egg bound by looking at, and massaging around her bottom area for an egg shaped lump.
A vet will probably be able to help the best, but if you want to try some 'at home' methods then you may be able to visually see a bulge, or feel that there is an egg there. If she is still walking about and relatively happy, then giving her some electrolytes will help hydrate and will also give calcium. Apply some lubricant to the area to help the egg exit easier and give her a nice warm area to relax for a while. If she is used to being handled and doesn’t find it too stressful then submerging her bottom area in warm water and gently massaging the area for about 30 minutes may help relax the muscles. An alternative to that would be a warm flannel held over the area. She needs to be kept warm and comfortable.
If the egg is physically too big for the pelvis then it will be impossible to encourage out and the only other option really is to collapse the egg by pricking it with a large syringe (large enough to pull in the contents of the egg). Afterwards let her relax on her own with fluids and calcium and she should eventually pass the remaining egg - it can take anywhere from an hour to a day. One risk with this method is that it is possible that a sharp edge of shell may cut the uterus. Usually the shell would collapse inwards so it shouldn't cut anything but it is a possibility.
If the cause of the egg becoming stuck is a small pelvis then it is almost certainly hereditary and prevention may not be possible unless it is the eggs themselves which are unusually large for that particular hen. Sometimes jumbo eggs can be the result of feeding too much meat/protein so if you feel that may be the case then cutting back on those elements may help minimize the chance of a large egg becoming stuck again. If the cause of the egg binding is a lack of calcium then crushed oyster shells may help (vitamin D is also needed to help the calcium be absorbed). In theory there should be enough calcium and the other vitamins and nutrients needed present in a balanced layers feed so supplements may not be necessary. If you do supplement and notice the egg shells getting pimples and/or pin holes appearing that may indicate that there is now too much calcium/vitamin D in the diet.
In the March 2011 Newsletter subscriber Stefanne Kelly gave her experience of an egg bound chicken which ended happily and includes photos of a jumbo egg with a smaller egg inside it and also an X-ray of her egg bound hen. Also in the same issue James Rousseaux shares his experience with an egg binding which sadly didn't end so happily and lists the symptoms and behaviour he noticed in the hope that the info "can save someone else's chook one day. Because knowing what I know now I could have saved our girl."