Look up Impulse. What you can get from Impulse is the average force. The average force is what breaks the egg. If you have the egg fall and hit the ground, it's average Force will be very large, it will have a large force spread out over a small amount of time. However, if you have the force spread out over a very long amount of time, then the average force will be small. If the average force on the egg is less than that required to break it's shell, it won't break.
To make a long average force, you need a long collision. For example, in a car, they are designed to accordion as the front end is crushed, that way it crushes in a /\/\/\/\ pattern, and it takes longer to crush. This makes the average force less, meaning less force experienced by you, and less injury caused. You want to do something similar with your egg. The longer you can spread out the 'slowing down' of the egg, the less the force and less chance it will have of breaking.
As for your design, that's up to you. I'm not an engineering major.