Hello
Sorry, nothing new for you but just wanted to comment that the name does not seem to have been overly common in Ireland in that timescale. PRONI Freeholders List of pre1840 contains no Muir surnames. At the following link, type in Muir, search will then show that the name was not that common
Surname distribution 1848-64. It also shows some possible variations of the name that may have been in use in the timescale. Interestingly, when the search is complete, if you type McIlroy in the second surname box on the results page, it then shows that in 1848 - 64 these two names only appeared together in Belfast (after your timescale, I know, but may help focus in on one part of Antrim as it were).
Have you had a look at
www.ancestryireland.com 1821 birth of an Alexander Muir just in case there is any connection to yours? Perhaps an earlier born child that died.
Finally, which part of Scotland did they end up in (I have recently been looking at a Muir link on one of my lines in Scotland)?