Green’s Dictionary of Slang

eager beaver adj.

(orig. Us) earnest, usu. excessively so.

1951		C. Stoker Thicker ’n Thieves 143: Sergeant Jackson, the police department’s eager beaver pimp and procurer.
1957		(con. 1944) A. Myrer Big War 332: Go ahead and pull it. You’ve been eager-beaver enough before.
1968		K. Brasselle Cannibals 136: You’re loaded with that eager beaver bullshit.
1986		F. Taylor Auf Wiedersehen Pet Two 133: They were commodities for the profit of the likes of Ally Fraser and this eager-beaver young man on the make.
2004		Webb & Lamb Secrets of Your Rising Sign 1: [...] with an eager beaver demeanor, they always do more than expected.

In derivatives

eager-beaverishness (n.)

(excessive) enthusiasm.

1963

F. Kohner Affairs of Gidget 69: I was all ready to go, but determined to control my eager-beaverishness.