Does anyone know why we think H-D is an egg? How about history of this nursery rhyme - how far back might it go?

Well in Through the Looking Glass an egg becomes Humpty-Dumpty. I’m not sure if that is the first connection between him and eggs however.

Cecil cites Martin Gardner in saying that “Humpty Dumpty” is a medieval riddling rhyme with the answer being an egg. The Annotated Mother Goose by William S. Baring-Gould seems to agree.

ignatzmouse, a slight correction: the article you refer to is a staff report by SDSTAFF Dex (aka C K Dexter Haven), not Cecil.

For a second there I thought Dex had been promoted.