In a comment on Must I, Then, Jim N posted that:

From a folksong book my grandfather gave me, I know that this is based on a German song, “Muss I Den”.

Here’s the german tune:

That’s definitely the same song. Good shot, Jim.

But what about the title? Google translate tells me that “Muss I denn” means “I have since” in English, not “Must I, then.” “Must I, then” must have become the title because it sounds the same. Maybe it was misheard by an English speaker who thought it was English.

And actually my favorite part of this song is the title. “Must I, Then” is a great name for a song.