Think like a porcupine
Or like a bee (a colony rather)
Only paper products and scotch tape. Too bad there are no straws or glue allowed
.
Wrap egg up in kleenex. Wrinkled not rolled, this shall give it a more collapsable 'honeycomb' effect. Wrap this in the tape so as that it won't come off of the egg. Wrap another wrinkly layer over that but use the tape sparingly, so as that it fits lossely.
Take sheets or paper and roll them along the diagonal to form a very narrow (thinner than a pencil) cone which is long (longer than a pencil by a good amount). It should resemble a spike. I suppose you could also use very thin card board strips (without rolling into a cone).
Then, tape the cones which we shall now refer to as spikes, to the kleenex wrapped egg. Don't tape too tight, but in a manner that will prevent it from moving more than a couple inches without tearing tissue. This will form a very spikey ball with very little air resistance.
Drop this... It should fall very fast since spikes of paper don't take up much room, and the kleenex wrapping has a negligible effect. Upon impact the spikes will bend and crumple, they will also sheer off of the kleenex, kleenex will 'unwrinkle' and perhaps tear, absorbing energy. All goes well and the egg will end up unscathed in a tangled mess of paper, tape, and kleenex.
To really impress your teacher... make it turn the energy of impact into electricity