Impulse is the change in momentum of an object. There are two impulses that your egg design will have to deal with. The first impulse is from the drop itself. Its momentum will increase due to the net force applied to it. The next impulse is from the landing. If...
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the egg vehicle doesn't bounce, then the two impulses in magnitude are equal!
Imagine that dropping from a meter, the egg has the whole meter to speed up to its speed right before impact (so the egg doesn't break just from falling because the force is small). At impact, that speed will decrease to zero (as long as it doesn't bounce). But, how much distance does it have when it hits compared to the meter? Not much, which means less time to deal with the same magnitude of impulse.
The basic formula for impulse is F*∆t. Since the impulse is constant, F and ∆t are inversely related to each other. More time means less force required to deal with the impulse.
So your design has two components. Reduce the overall impulse the egg has to end up dealing with, then reduce the force of impact.
Since the egg is falling, it would be best to have the egg take more time during the fall, which means that your net_force is smaller than that of just the weight of your vehicle. Adding an opposing force (drag from a parachute - not bouyancy as suggested by other posters - bouyancy is due to differences of pressure - unless you are using helium balloons, the effect of bouyancy is negligible) makes your net_force smaller. Reducing your overall impulse.
-Wait: won't the decrease in force increase the time to fall, thereby keeping the impulse the same? No, but good question. Why? Let's look at a simple case to see: if x=5 meter and a =10 m/s^2 (free fall), then it will take 1 second to fall (impact at 10m/s). Let's call this impulse J. If we halve the net force, then a = 5 m/s^2 and it will take 1.4 s (impact speed of 7m/s). So the result is an impulse that is 0.7 J definitely smaller!
Second component: reduce impact force means increase impact time. Cars are built to crumple, which increases the time of impact greatly. So your design should absorb and deflect the force of impact.
Note that the egg can handle pretty large impulses (you can throw an egg like a baseball, and sometimes eggs won't break when they land from a very high toss to grass). It's the force and distribution of that force that breaks the egg. (i.e. a force focused at a point vs. the same force distributed over a larger area).
I think you can be creative with the materials given - so I won't tell you what a good design might be - by understanding what impulse is and how you can manipulate the time to fall and the time of impact is key to a successful design.