Step 1: Concept
First of all, the arm must be able to pick up the egg without dropping it. Secondly, the egg must not break when it is picked up. As a result, there should be a range of forces we can use on the egg.
The design of the arm will consist of two four-bar linkages for the two sides of the gripper, with a piston supplying the force. The head of the gripper will be coated in rubber.