In Paternal Egg-Stinct, Marlene found an abandoned egg in her habitat. She gave it the penguins to look after, since the incubation process is technically a "guy-thing" for penguins. Julien wanted the egg for himself. He called it Julien Junior, or JJ for short. Private called the egg Eggy.
At the end, the egg hatches. It's a duckling. Marlene brings Mother Duck to the boys to get her duckling back.
Unfortunately, the effects of the Penguins' training goes to a point where Momma Duck asks the Penguins to do something to stop him (see Hard Boiled Eggy). Their attempts to show Eggy the dangerous portion of their missions fails, as the duckling goes on a rampage taking down all the animals in the zoo, Eggy inherits 5 traits. In the end, Julien teaches Eggy how to dance and Eggy decides to adopt Julien's mannersims instead of the penguins' mannerisms.
Because Eggy was born under the Penguin's watch, he has inherited individual traits of the penguin's personalities: Skipper's gusto, Rico's love for destruction, Kowalski's knowledge, and Private's adorability. After Hard Boiled Eggy, he gains one more trait: King Julien's enthusiasm (and possibly his accent, as he speaks like this in Cute-Astrophe and Siege the Day).