Does it bring back fond memories? It does for me. It seems to be late-19th or early 20th C., and in the sixth row, right tend, we see wool being dyed right on the farm. So when the singer says,
I'll dye my petticoat, dye it red...
it may not be as far-fetched as it seems. ========== I don't believe red has any special meaning. It simply rhymes with bread and dead.