... I have found the death certificate for Agnes Ann McCulloch who died in Saltcoats on the 28th of Jan 1879 and lists her parents as James Beggs (Mason, journeyman) and mother Sarah Beggs (nee Elliot)
It's also worth noting that the Thomas Beggs I mentioned at reply #5, ie. the son of James Beggs and Sarah Elliot, went on to marry an Agnes Cain in 1867 in Kilbirnie, about 10 miles northeast of Saltcoats, Ayrshire. I'm reasonably comfortable that it's him, because both parents' names are correctly mentioned and James' occupation is 'Mason'. A Samuel Beggs was one of the witnesses. I'm also comfortable that Thomas ended up in Seattle, King County, Washington, USA, where he died in 1924, for there's a death record that mentions both parents' names and other (eg. census) records to corroborate the move to the USA.
BTW, here's his grave:
https://www.findagrave.com/memorial/59583351/thomas-beggs