"Visible from all directions (open structure)", and "can not be built into the structure" are poorly defined conditions. I
guess that first condition means no more than roughly half the egg can be cupped or otherwise obscured by any form of hammock or restraint . It's pretty hard to mount an egg to a vehicle without making part of it invisible. The other condition,
I guess means the egg may be easily mounted, by your teacher.
It is pretty easy to build a vehicle to cushion simple drop - the challenge is where that rebounds and flips over on its side after initial impact. Fail. So a lot of these are inelegant bouncy balls/cages. Which is overkill, bad engineering, unimaginative in my humble opinion.
I think your balsa wood cage safe enough. Popsicle sticks or the like will be easy to build with, maybe just soda straws. Use string, lots of string, to tension that together and also hammock the egg.
30cm cube max? There is a nasty way to do this. Get yourself a load of egg cartons, yeah 30cm cube of them glued together, with the egg in top seat. And

fill the bottom cartons with sacrificial eggs.

Splat. Ask your teacher if the egg may rest in an open egg carton - that'll illuminate her conditions somewhat and maybe you could build something less bombastic out of egg cartons.
If you can drop onto soil (not pavement) then something like a lawn dart might do well.