Oooh. I did this in, um, sixth grade, I think. It was fun! We were required to only use tape, straws, and I think that was about it. Maybe one or two more things. Anyhow, you know those big round straws? (If you've ever seen those Pearl Milk Teas, those straws are the ones that go into those teas. Ick.) Well, those are really useful. You can cut up the straws so they're almost like packing material and make a bottom that will support the egg. And remember to put some more straws on the top to protect it. That's what I did, at least. Be careful, though... my first try bombed, and my second try worked like a charm. So try different things. I'm not sure what restrictions or what loopholes you can use, so...

Incidentally, if you're allowed to, there was another group who made a parachute out of tape, which was the coolest thing ever. I don't remember if it worked, though. =)

I hope that helped. Good luck constructing your container!

~Crystal109