Answer
We can minimize the force that the hands exert on the egg by moving the hands back together with the egg and trying to remove the egg's momentum as gradually as possible over the longest possible time.
Work Step by Step
When the egg reaches the hands, the egg has a certain amount of momentum. The force of the hands catching the egg exert an impulse on the egg which removes this momentum. In order to remove the egg's momentum with the least possible force, the hands should maximize the length of time that the collision takes place. We can do this by moving the hands back together with the egg and trying to remove the egg's momentum as gradually as possible over the longest possible time.
