A similar game is...
Wofa, you wouldn't happen to know anything about this, would you...?

Sorry, I know the Punch-buggy Red (or whatever color) game but not the hit-the-rodent one.

"Beaver" can have many meanings, of course, ranging from the lascivious to the bland. Even without resorting to a dictionary we can give it implications of hard porn, or soft porn, but I would guess the relation to fake veneer most likely arises from the belief that a beaver fells its wooden "victim" by scraping off thin layers, and this is what's made into the veneer facing. (Just remember, though, this is the Premarin-guesser speaking!)

Punch-buggy-* has the potential to have a resurgence now that VW has hatched out the new Bug. Just think of it - a whole new generation of black-and-blue arms! (I always preferred the "padiddle" game, myself - you know, first to spot and proclaim the oncoming car with one burnt-out headlight gets to kiss the companion.)