Is it a chicken egg? Duck egg? Frog egg? Snake egg?
It doesn't really matter I guess. In my opinion, there are 4 parameters to consider first.
1. The material of the floor onto which the egg will be dropped on (critical)
2. The height at which the egg will be dropped from (not that critical)
3. The mass of the egg + shell
4. The strength of the egg shell
The egg breaks easily when dropped because it receives a 'point' load, ie. all impact force is transmitted to the egg through a very small area. If you put an egg on a flat surface, you can see a very small area that egg comes in contact with the flat surface.
The height of the egg gives the egg a potential energy of Ep = mgh where m=mass, g=earth's gravitational constant, and h=height. Upon impact, this energy has to be dissipated. When the egg hits a very soft surface (like mattress), the movement of the mattress absorbs the impact and protects the egg (also, there is no pinpoint load on the egg -> the load is now somewhat distributed over the much wider area!)
Therefore, at the point of impact on any hard surface, the egg will have to take up much of the energy (because the hard surface will not going to give). The egg cannot dissipate this energy by flexing its shell (it is rather brittle). Combine this with the momentum of the eggy fluid that is trying to burst out of the egg, and you have broken egg and a mess.
When we handle egg, our palm is soft and does not at any time apply a point load. However, pretend that you are wearing a glove with sharp needles. The egg will not last long. Do an experiment. When you squeeze an egg with your thumb and index finger on its long axis, it will take quite a bit of load. Now tape tacks on your thumb and index finger (pointy thing away from the skin of course) and squeeze the egg using the same pressure. Bingo!
I wish I could give more but I have to go. I hope it helped and good luck in your search!!!