Remember, the point is for your egg to survive, not your structure. Build something that will break as it hits the ground, slowing the fall and saving the egg. Zig-zagging your popsicle sticks and using a little hot glue (not enough to make it very strong) will result in a structure that snaps as each piece hits the ground, but the brute force of impact is therefore distributed throughout the machine. If your egg is in a stable structure at the top (perhaps fins of some kind out of popsicle sticks to keep it upright as it falls?) it should survive unscathed.
In that same vein, remember that wide flat surfaces are less likely to hurt the egg than, say, thin toothpicks which might cut into it. Conversely, toothpicks can bend a little, whereas flat surfaces of wood will just hit the egg. Your call to find a happy medium!
Source:
We all did physics class back in high school, and I can't believe I revived my high school email account to answer a question!
Oh, and my egg survived back in the day.