I just realized that John Lennon's "Happy XMas" is the same tune as the old folk standard "Stewball". And I just learned that the name of the horse, and therefore the name of the song, is "Stewball", not "Screwball". I've been singing it wrong, to myself, incorrectly, for ~40 years. I like "Screwball" better.

“Screwball” is nice. “Stewball” and “Skewball” are both in versions of the song going way back, I think “Skewball” was the original name.(because the horse is skewbald).
“Screwball” just seems like an appropriate name for a horse that only drank wine.
the Google hints that perhaps Roger Miller did a version called “Screwball”, but i haven’t been able to confirm.
Another song about Skewball