Many, many RC planes are made out of balsa wood. There are many internet resources on building them, as well as many hobbies shops around that sell RC plane kits.
Start by doing google searches on RC Airplanes, Airplane models, etc. Wikipedia, Answers.com, etc. are other great places to search.
The bigger the plane, the easier it will be to carry the weight of an egg.
You also have a second problem - having the egg survive intact. Lots of schools, and engineering programs have egg drop contests. Do a search on those.
If your plane can land on its wheels, or just land in a somewhat controlled fashion, the egg will easily survive. Your challenge might be to protect the egg if the plane doesn't land elegantly.

There are many ways to do this. Some quite ingenious, some basic, some complicated, and some hillarious. All pretty elegent engineering. Search and you will find these. Weight and drag will be your enemies. Lift your friend.
I'm assuming that you really need to build a glider, and the plane will not be self powered. Therefore, you will probably be dropping it from some height. It that true? Wing design is important for stability. You will run into terms such as dihedral, lift, and flat-bottomed vs. symmetrical wings.
Placing the egg in the center of gravity will promote stability.
Does the plane even have to fly, or can it just look like a plane and roll? Does it even have to look like a plane.
So many unanswered questions.