Summarize:

Suspend the egg in the center of a box or similar container using a pair of nylon stockings or panty hose. Stockings are very elastic and soft. As the container holding the egg strikes to ground, the stockings should give slightly, allowing the egg to come to a stop without that stop being quite so sudden. As a result, the force on the shell is lessened, making it less likely to crack.  Cut off one leg of the stockings. Place the egg inside this leg at the center of stocking. Tie the egg in place using rubber bands. Tightly pull the stocking leg diagonally through a box, extending it from one upper corner to one bottom corner. The egg should be positioned in the center of the box. Staple or otherwise hold the stocking in place. Note that your box can be made of nearly any material. It could be a cardboard or plastic box, or you could even construct a box frame out of metal clothes hangers. You can place the egg at the top of a cushioned container instead of placing it in the center as long as you have a heavy enough weight to control the direction the container falls in. The easiest way to do this is to use a stone and styrofoam cups.  Place one heavy rock in the bottom of one styrofoam cup. The rock must be heavier than egg. Place six more styrofoam cups inside the bottom cup, on top of the rock. Place the egg inside the top cup. Fit one more cup gently on top of the egg to secure the egg's position. Tape the cups together along the sides so that the container will not fall apart during the fall. If the rock is heavy enough, the container should fall rock-side down and egg-side up. The styrofoam cups should also help cushion the blow. If you can design a parachute for the container holding your egg, you can reduce the velocity at which the egg drops. Since the egg is falling at a slower speed, the force of the impact will be much less once it hits the ground. Less force means that your egg might have a chance at survival.  There are a few different parachutes you can try, but one of the easiest materials to use is a plastic shopping bag. Place the egg in a box, along with any cushioning you may choose to use. Attach a large plastic grocery bag to the top of the box using tape or a stapler. Make sure that the handles are near the sides of the box so that enough air can get inside the bag as the box drops. When you drop the box, make sure that the side you attach the parachute to is at the top. This will allow air to fill the bag and cause it to open up, reducing the speed of the descent as a result.
Create a cradle. Weigh down the bottom of your container. Make a parachute.