In this blog (really the video) I will explain how to build a completely full auto egg farm. Now first, lets define full auto. This can mean several things for chickens. You could cook them, or get eggs off of them. This is more for the eggs. Now be warned before I start, this will cause exponential amounts of lag, so if you have a crappy computer, please keep it toggled off most of the time for your own sake. Also low power servers might not be able to handle it.
Essentially how this works, is that chickens lay eggs into a hopper. The hopper puts eggs into a chest. Another hopper then takes the eggs out of the chest. I did make this farm toggle-able so you can go in and get eggs. That hopper feeds into another hopper which feeds into a series of droppers. The droppers feed into a dispenser which throws eggs. The eggs hatch into chickens. The chickens then grow up and lay eggs themselves. Thereby creating an exponential amount of chickens. (Of coarse, once you get tens of thousands of chickens, the system will no longer be able to throw the eggs as fast as the chickens lay them. Your game would have long ago lagged out though. So unless you are playing on a supercomputer, this is an invalid concern.)