DS did this last semester in his physics class. They used raw eggs, supplied by the teacher so no one could sneak in a hard-boiled ringer.
DS used a styrofoam ball, either 4 or 6 inches. He cut it in half and hollowed out some of the styrofoam in the center. Then he stuffed both halves with cotton balls. He enclosed the egg in a ziploc bag, placed it in the center of the cotton, and then closed the ball with tape (duct tape or electrical tape, I think). He cut a couple of plastic straws into short pieces and stuck them into the exterior of the ball.
His class dropped their egg carriers from about 25 feet up onto an asphalt parking lot. His egg came through intact. They had to do two drops with the same carrier, and he had a little scare on the second one - the tape came loose and the egg rolled out, but didn't break, fortunately.
He took digital pictures of his assembly and test drops, including some designs that failed miserably - messy!

He included those in the report that he had to turn in.
His older brother had the same teacher and same project four years ago. I seem to recall someone used a round loaf of bread with a hollow in the center for the egg.
Good luck!
