I found this way after you posted, but my grandfather sings a song like this:

Mr. Moon, moon, bright and shiny moon,
Won't you please shine down on me
Mr. Moon, moon, bright and shiny moon
Come out from behind that tree

For there's a man out there with a big shotgun
He's out to get you so you bound to run
Oh Mr. Moon, moon bright and shiny moon
Won't you please shine down on me.

My grandfather is from the south, so it's unlikely that in his version it's "Johnny Reb" He was born in 1903, so it could be an old WWI era song

I too have been unable to find any other references to the history of this song.

Elizabeth