Hiya Ian,
You've not mentioned the townland in which the Muir family lived. Griffiths Valuation of Co. Antrim which was conducted in 1861-2 lists the following in the Parish of Shankill in Belfast ...
William Muir, St. Ann's Ward, Townparks, Thompson's Court
William Muir, St. George's Ward, Townparks, Lemon Street
Spears Muir, St. Ann's Ward, Townparks, Townsend Street
William Muir, St. Ann's Ward, Townparks, Thompson's Court
William Muir, St. George's Ward, Townparks, Lemon Street
Spears Muir, St. Ann's Ward, Townparks, Townsend Street
There were a large number of McIlroys in the Parish of Shankill.
The Belfast section of the 1852 Directory of Belfast and Province of Ulster Directory lists ...
Speirs Muir, bookbinder, 18, Boundary Street
There are ten listings for McIlroy in the 1852 Directory. There'd be no harm looking at earlier directories.
Christopher