lmao @ both Dullard & Gayner.
OP: Suggestions - if you have a shrink wrap machine - hell, just put the egg in a ziplock bag and get all the air out of it (submerge it.) Cut the base off a 2-liter bottle, and cut it off near the top - keep it longer than 5" for now, you can trim it later.
Center the egg (inside the plastic bag) in the center of the 2-liter bottle and tape the edges of the bag to opposite sides of the 2-liter bottle. Put something over the bottom of the bottle & fill it with Great Stuff foam insulation.
Wait for the insulation to dry. Then, with a sharp utility knife, carefully cut down the opposite edges of the bottle where the plastic is. You should be able to split the whole thing into two halves, revealing the egg in the center of a perfect fitting coccoon. You could use this coccoon and a couple rubber bands to hold it together for the drop.
Major tip: not all brands of 2 liter bottles are identical in size, although it looks like they are. One brand's bottles will just barely slide into another brand's bottles when you've cut them off. (Sorry, off the top of my head, I don't remember which goes into which, but I *do* recall that you can find that info on Google.) Then, you can simply sandwich the egg between the two form fitting pieces of foam, and slide the whole thing into another bottle which will help put a little compression on the whole thing.
fwiw, when testing this method, prior to cutting open a bottle to unencase the egg from inside its shrinkwrap bag, the egg survived being spiked on the floor & survived being hit with a baseball bat.
Another method, look up "ooglick" or something like that. You want to create a very dilatent fluid - one which under different forces will turn into a solid. Suspend the egg in the center of such a fluid & it's like having it encased in concrete when it hits the ground. The pressure is relatively uniform around the egg, and it doesn't break. One material that works quite well for this is a very thick slurry made from corn starch. I saw an egg survive a roughly 100mph+ impact using this method. (My team's device in an engineering competition. Unfortunately, the other half of our device malfunctioned. Rather than launching our egg a few hundred feet in the air, our launching device when haywire & decided that rather than go up, it was going to follow a roughly parabolic path into the ground at max speed. Points were based on time + if our egg survived. I believe we set a record for length of path in the least time with a surviving egg.