I have given the infamous egg drop assignment a number of times, and I have seen some definite trends in successful attempts.
First, with two eggs it's important to separate them from one another in the container. Second, you should use an inner packaging material that will keep the eggs cushioned and prevent them from moving around. This could be Styrofoam packing material, shredded paper, or something similar.
The inner package should be enclosed in an outer package that is designed to crush or crumple. The idea is to let the outer package disperse the energy of the impact by deforming, protecting the inner package and its contents. An alternative is to create an outer package that will increase air resistance and slow the fall rate; a parachute or something similar can help give your eggs a soft landing.
One of the best products on the market for cushioning is the insulation foam that comes in the spray cans; "Great Stuff" is one brand. However it is really sticky to work with; if you try it, read the directions carefully before your start.