My understanding of linen making is that in the early 1800s it was mostly a home based industry. The larger commercial linen mills didn’t really get going till the mid 1800s. (The necessary machinery wasn’t invented till then). See this link for linen making history:
http://www.fergusonsirishlinen.com/pages/index.asp?title2=History-of-Irish-Linen&title1=About-Linen
If your ancestors were making linen in the early 1800s I’d guess it was home weaving (which every farm labourer did, especially in the winter months). If the family did have a mill it was more likely a corn mill or such like.
You could search PRONI's records to see if they have any record of a Gillis mill:
http://www.proni.gov.uk/index/search_the_archives/ecatalogue.htm