Looking for Cockburn S IN U S a Canada

I am looking for the son of my ggrandmother Margaret McCabe. She had an illegitimate son called James. My ggrandmother married Thomas Cockburn & they came to Australia,from Scotland early 1900's, James who had the surname Cockburn stayed in Scotland as he was serving with the Black Watch. After discharge he went to either Michigan or St. Thomas Canada. Any help with this would be appreciated.