Bridgett,
The surname Aiken/Aitken is fairly common in Antrim & Down. In the 1901 census there were 280, 15 of which were named John. It would have been equally prevalent in 1807.
Methodism didn’t exist as a separate denomination in Ireland in 1807. Methodists didn’t break away from the Church of Ireland till 1816 at the earliest and many continued to use it for baptisms and marriages up to the 1870s. So for someone born in 1807, who later became Methodist, you might need to look at Church of Ireland records. (There are no Methodist records to consult).
The 280 Aikens in the 1901 census of Antrim & Down were a mix of Presbyterian, Church of Ireland and a few Catholics.
If the family perceived themselves as Scots-Irish then the most likely denomination is Presbyterian, though they could be Church of Ireland too. Half the population of Counties Antrim & Down are descended from Scots who moved to Ireland in the 1600s, so that aspect of the family history is probably sound.
Not all churches have records for the early 1800s and of those that do not all are on-line, so you shouldn't assume that the birth you found in Lisburn is the right one. There will have been many more but just those records are either not on-line or are lost. (A lot of early Church of Ireland records were lost in the 1922 fire in Dublin).
Without parents names from a marriage or death certificate in the US it’s going to be difficult to be sure you have found the right family. And without some idea of where in Antrim or Down the family lived, it’s a bit of a needle in a haystack.
Possibly DNA testing may be a way of matching with others who have additional information about where the family originate. Family Tree DNA reportedly has more people with Ulster roots than any other company. That obviously increases the chances of finding a match. You might want to try them or, if you have already tested, you can transfer your results to them for no fee.
The North of Ireland Family History Society is running an Ulster DNA project in conjunction with FTDNA and can offer testing kits at a reduced price. http://www.nifhs.org (Go to DNA project on the website).