EntropicBliss said:
You could also do a parachute with weighed prongs on the bottom, and cushions between the egg and the prongs. Think of a lunar lander with a parachute.
I tried this general design once, but the restrictions were a lot more dire: only toothpicks and glue allowed. Without a weight to orient the "lander" though, the meshed toothpick shock-absorbers didn't have the chance to absorb the blow, and it simply landed on its side, leading to cracked-egg syndrome of course. Weaker side shock absorbers couldn't handle the full load.
Actually, nobody's design succeeded. Toothpicks are terrible shock absorbers, and there's no way to properly orient such a structure without weights or MASSIVE quantities of toothpicks.
More OT: Essentially anything soft is your friend. Fluffy things which contain lots of air, such as popcorn, packing peanuts and bubble wrap as mentioned above are your friends, but keep in mind you may need a lot of it.
The goal is to decrease the amount of acceleration the egg experiences when it hits the ground, and that means
soft. A pringles can or other similar edged device to contain the egg on its fluffy pillow of material is also a good idea, because the force of impact will by translated into the cardboard edges. Failure to weight properly will mean disaster.
Keep in mind as well that the structurally strongest part of the egg is the pointy bit, so you'll want to have that facing down in your final design.