1. Paul just discovered jogging as a form of exercise, and he went out and bought new running shoes, a new tracksuit, and all the other gear that goes along with it. He’s a real eager beaver about jogging.
2. Dieting must be done slowly and carefully. Don’t be such an eager beaver to lose weight that you harm yourself by not eating anything.

The expression suggests the image of an anxiously working beaver, which is reputed to be an active, industrious animal. It has a slightly negative connotation, as of someone eager to impress others with his/her effort.