In an egg drop, we observe the egg in free-fall. If we draw the egg in a free-body diagram, we can observe that the only force acting on the egg is gravity (assuming no air resistance). As such, the acceleration of the egg is 9.81 m/s^2 [down].
This relates to Newton's First Law, which states: "Unless an unbalanced, external force acts on an object, it will maintain its state of rest or of constant velocity. In short, if Fnet = 0, then a = 0". In this case, there is an unbalanced external force acting on the object: gravity. Because of this unbalanced force, the egg accelerates downward.