I grew up in Memphis and this is how I personally heard the song for the game goes:
Down by the river of hanky Panky
Bullfrog jumps from bank to bank he
Says "ooh! Ah! Skittle skat! Big fat dirty rat!"
I pledge allegiance to the flag, Michael Jackson is so bad
Coca-cola burnin' up, now it's time for seven up
Seven up has no caffeine, now it's time for billy Jean
Billy Jean went out of town riding on a pony
Stuck a feather in his head and called it macaroni
Mickey Mouse had a house
Donald Duck messed it up
Who will pay the con-se-quenc-es
Y-O-U

And the person who's hand gets hit on the last U is the "loser" and is removed from the circle and we keep going with the game