Paddy Mac, you have been trying to get an answer for some time, I see by past threads. Not an easy one to solve. Quite a bit in threads 7080 and (especially) 17359.
Fag an Bealac, 25 June 00, posted lyrics from the 1860s (dating not completely clear- what was the date on the printing of the Beadles Song Book for the Irish Jaunting Car? (Valentine Vousden). Note that in the song, "on the door is a star."
The Bonnie Blue Flag was published in 1861 (Harry McCarthy). It contains the words "bears a single star."
The melodies are very similar, and the lyrics posted by Fag an Bealac and those for Bonny Blue Flag (see DT) both go well with the melody.

Which came first? Both were popular. There seems to be no clear evidence of which came first. McCarthy is listed as an Irish immigrant; did he take the Irish melody? Or was it the other way around? The Bonny Blue Flag was first used in a revue in New Orleans in 1861.

Thread 17359: Jaunting Car
See the DT lyrics for Bonny Blue Flag and thread 33188: Bonny Blue Flag