The Facebook article says, "In the year 1878 when Lewis Potter was Jailed, J Clelland, Chairman of the Cunard Steamship Company bought the mansion and owned it for over 20 years. The last recorded tenants in the house were the Burns family ...."
The error is that it was James Clelland Burns, not J Clelland, who bought the house and lived in it. I assume that they were already renting it in 1871 from Lewis Potter; that in 1881, having bought it three years earlier, they let it to William Connell while they were living in London; and that they returned to live in it by 1891.
According to the University of Strathclyde Archives web site Burns & Laird Lines Ltd, a subsidiary of Coast Lines, was formed in 1922 from the amalgamation of Laird Line and G & J Burns, two long established Glasgow companies which had pioneered steam services between Scotland and Ireland. The new company offered freight and passenger services between Scotland and Ireland. The company ceased to exist in the 1970s. Therefore the Facebook article is correct in saying that the Burns family who lived in Glenlee House were connected to the Burns Laird Shipping Company, but that company was clearly formed after the death of James Clelland Burns in 1908, and after the family had ceased to live at Glenlee House.
During the lifetime of James Clelland Burns, the Burns family owned and ran the shipping company G and J Burns. Their services to the Western Isles were sold to their nephew David MacBrayne and eventually became Caledonian MacBrayne. Many of their other services were amalgamated with various other companies that eventually became the Cunard shipping company.
It was Sir George Burns, and later his son John Burns, later Lord Inverclyde, not J Clelland (or even James Clelland Burns, the occupant of Glenlee House), who were the heid yins in the Cunard company.