The most basic form of this exercise lets the students take a run at designing a protective case for dropping eggs from varying heights. The goal, of course, is to have the students cover their egg so that it doesn't break when dropped. For larger classes, organize the students into teams of three and give them parameters for what materials to use. You can also allow the students to pick from materials of their own choice, but be sure to specify what materials won't be allowed (glass, metal, etc.). Give the students a week to design their crash cars and then spend a class period dropping them to see whose does the best.
Have the students design more of a lightweight crash car to increase aerodynamic properties. Instead of dropping the eggs, toss the eggs or roll the eggs (for crash cars with wheels) from a fair distance at a nearby wall. Make students aware that their eggs must be designed to reach the wall first and foremost. This can be done in teams or individually depending on class size. Throw or roll the eggs at a wall to see which students designed the crash car that doesn't break but makes it to the wall.
Suggest to students the changing properties of eggs based on temperature and the addition of chemical compounds. This would be ideal for older students in a high school chemistry class. Instead of designing a crash car, show them how to freeze eggs, boil them or coat them in different hardening or softening chemicals (i.e. rubber cement, spackle or ice). Allow each student to pick a method or coating they think will work best (or look interesting when destroyed) and then test by dropping them out of a window on to concrete or grass.
Allow the students, in teams of two, to decorate, freeze and possibly coat their eggs in teams. Each team should be looking to create hardened eggs with distinctive markings so that they can be found among one another. Then have them set up a baseball-like format where one student pitches to the other and hits the egg with a baseball bat or tennis racket into an open field. Leave the eggs where they land until each team has hit. Have one member of each team blindfolded while the other member gives them directions to the location of the egg in the field. The blindfolded team member must retrieve the egg still intact for their team to win.