When we did this, the one that worked best was modeling clay - surround the egg, and wrap with cardboard.
Around here – and I had thought in most of the U.S. – schools typically let out in late May. This is public, private, and parochial.
This kind of thing varies a lot from place to place, though. And the kids here go back to school mid-August … maybe in other locales, they start later.
I won this contest in 7th grade by putting the egg in a semirigid plastic cup, like the kind they give away with kid meals at Chilis/Fridays/etc., full of peanut butter.
Boy, times have changed. We finished school the last week of May/first week of June and did not have to report back until after Labor Day. Of course, that was when going to/from school was uphill both ways, snow/cold didn’t stop us (and school didn’t close), and Mom/Pop/school buses were for sissies. Real school kids walked or rode bicycles.
Oh yeah. The new kid in the fall who came from another state often was held back a year cuz our state standards were tougher than 48 of the 50 states.
And teachers used RED PENCILS to mark tests and papers.
Kids got it easy these days.
How about this:
Take a roll of paper towels. Find an egg that fits snugly inside the cardboard center. (Large eggs fit almost perfectly. Nice and snug.) Stick the egg in one end. Stuff the other end loosely with a few paper towels. Put a weight on the opposite end from the egg, so that end lands first.
Drop it.
The roll should land on the end opposite the egg. Friction between the egg’s sides and the inside of the cardboard roll will slow the egg’s descent through the tube upon impact, and the paper towel wadding will provide additional braking. When the roll then falls over on its side, the paper towels on the roll itself will absorb that impact.
You might even get this to work using just the cardboard center and a few paper towels for stuffing. Maybe a little padding on the outside.
Adding the weight would be key, to make sure the roll lands on the correct end, opposite the egg, and also to prevent the contraption from bouncing too high on first impact.
Is there a weight or size limit? Seems to me I recall that the Duke competition (mentioned by Attack from the 3rd dimension) had some limits of this type.
I’m getting flashbacks from Jonathan Swift here, but ***which end of the egg is stronger ? **[/I
This would seem to be a crucial preliminary question.
My favorite egg-drop thingy, which shouldn’t be replicated as it shouldn’t have worked, was an ice cream cone made of drinking straws witht he egg where the ice cream would go.
I thought that the cone would fall straight down, and the point of the cone would keep the egg from breaking. Of course, because the egg was so much heavier than the drinking straws, the cone rotated in mid air.
But it didn’t flip all the way around. No, by the time it hit the ground the cone had only turned half way, with the side of the cone almost parallel to the ground.
Instead of the side of the egg hitting the floor and breaking, the cone hit just before the egg did and wedged the egg out of the cone, sending it skidding across the floor totally unharmed.
I did a shock suspended egg solution when I had to do that, of course it was 5th grade.
I built a padded little egg shaped cage with eight holes drilled for key rings. The found a cardboard box considerably bigger, poked holes in the corners and threaded rubber bands through the key rings and out the holes and tied with good tension.
Worked like a charm, plus it’s kind of cool when you drop a box that hit the ground, sits still a fraction of a second then makes a little hop. 
Why wouldn’t you just wrap it in bubble wrap 40 times? Or seal it in the center of a cubic yard of polyurethane foam in an egg shaped cavity? I’m not very good at thinking like a first grader, but if I had a first grader with an egg to protect I’d do one of those two.
In a team-building exercise at my last job we were given the egg, 2 plastic drinking straws and about a foot of sticky tape, with a drop height of 6 feet onto a tile floor. My team was the only successful one of 4. My idea was accepted by the team. We (allowably) cut the straws into short lengths and taped them in 3-4 layers to the Brobdignagian end and it worked.
Nobody’s mentioned the obvious yet? Put it in a 15 foot tall container, so when you drop it it hits the ground immediately.
Hey, don’t look at me like that! I’m a theorist.
Better yet, put a rocket engine underneath the egg to slow it’s descent.
This is almost exactly what I did in school, as well. Worked every time.
What I did when we had this assignment in high school was to take one ziplock sandwich bag, inflate it maybe 75-80% with air and seal it, rest the egg on top and stack another loosely inflated ziplock bag on top of it, both of them taped around the egg. Then I opened the top one a bit so the force traveling through the whole thing on impact would waste energy blowing air out the top bag. Now that I think about it, I wonder if I couldn’t just leave the top bag off entirely.
Now the fun random thing that I learned from a C-130 Loadmaster in the Texas Air National Guard. They can air-drop fresh eggs from several thousand feet up. They use a special kind of pallet where you have a packing material not unlike corrugated cardboard, with a layer of eggs, then another layer of the cardboard, and so on. It was designed so that, when the pallet landed (parachute assisted, unless somebody messed up rigging the thing), the force of the impact would collapse the cardboard layers, but they would in the process cushion the landing of the eggs. Most of the eggs would survive the impact (a few would be broken, but then you can’t make an omelette without breaking a few eggs!)
I clicked on the thread to suggest this.
If you do this, then you need to drop several dozen eggs at once to simulate a StarShip Troopers landing scene.
Eons ago I checked out a model rocketry book where some kid did exactly that with model rocket engines 
It would work, but I’d imagine like all things rocket its pretty hard to get things precise enough to work reliably.
I like the paper towel roll idea myself. Put fins on it. Also put a big blob of clay on the bottom end so it fall right end down. Also that clay will absord a good deal of the impact and minimizing any bouncing that might occur.
Best strategy I’ve ever seen uses plastic wrap. Cut a long piece, and roll the egg across the short side so that there are several layers around the egg (or use two pieces, staggered, just to make sure). The long ends will dangle on either end of the egg. Now cut two holes on opposite sides of a rigid container, like a popcorn tin. Pass an end of the plastic through each hole, and secure firmly, leaving the egg suspended in the middle of the container. Drop it from 15 feet – hell, 30 feet – and no harm will come to the egg.
Attach it to a helium balloon, filled to just under neutral bouyancy.
This is why I’m guessing the competition has size and weight limits.