Seems to me the egg-drop has two issues:
1. Assuring uniform distribution of force around the "bottom" of the eggshell.
2. Reducing the g-loading that the egg sees.
For (1), I'd guess that the hot ticket would be spray foam from Home Depot, perhaps the Latex version. Hold the egg in your container, spray foam around it, and you have an excellent, form-fitting, force-distributing holder. If the judges don't like foam sticking to the egg after the event, wrap the egg in plastic wrap first.
For (2), I'd suggest using the "crush zone" principle. The greater the distance that it takes to slow down, the smaller the acceleration experienced. Use a rocket shape with trailing fins to keep the nose pointed at the ground. Place the egg just in front of the center of pressure so the rocket is stable. Fill the front half of the rocket with a crushable, non-rebounding material - tissue paper, perhaps water contained within a thin layer of plastic designed to burst on impact, even cardboard folded appropriately. The device should fall, crush, stop, and fall over - no rebounding! You will probably have to do some experimentation to get the right characteristics out of whatever material you use, but it shouldn't be difficult.