Neil is a rather unusual name and might be given to other relatives or might indicate a wife's surname.I would check for any Neil McCLearys/McLearys/McCleerys in the County Antrim records especially near Carnmoney if the family is Catholic. McClurys, too, if the handwriting is bad.There are some 1851 census records on line for some parts of Antrim containing Catholic McClearys.
If not Catholic, Belfast and Killyleagh are major McCLeary centers. They have a better chance of being in PRONI free on-line freeholder records.I have never heard of a Neil in the Armagh/Co. Down border area, another center.