Of course the song is more about alcoholism than weaving!

7000 weavers went on strike on 30 June 1787 and stayed out until the men were killed on 3 Sept same year. I found another two names of the dead: Alexander Miller and James Ainsley. They were protesting against wage cuts. Calton had been chosen as a weaving village in the first place because it was outside Glasgow and therefore beyong the influence of the weaver's craft guild (early trade union).
The village's original name when founded in 1705 was Blackfaulds but it was renamed Calton in 1723. Nobody is quite sure of the name's derivation, best guess is from 'cauld', or weir, giving a good supply of water for early industry. Of course there is also a Calton Hill in Edinburgh but the Glasgow one is the more probably as it was a weaving community.

There's a decent outline history of the Calton in a book called 'Villages of Glasgow' (volume I) by Aileen Smart, 1988, from which much of the info posted here by me has been taken.