Brittany,
McDavid is not a common name in Ireland. Looking at the 1901 census there are less than 20 of them, mostly in or near Co Antrim which ties in with your family background. Most were Presbyterian, indicating Scottish origins. In the 1700s (which is when you have said your ancestors left Ireland) the vast majority of migrants from Ireland to North America were Presbyterians (native Irish didn’t start migrating in large numbers till the 1800s). That and the general location (ie Co. Antrim where so many Scots settled) points to them being descendants of Scots who arrived in Ireland in the 1600s. In North America they are known as Scotch-Irish. In Ireland we tend to call them Ulster-Scots.
The other family name you have (Mattison) also sounds Scottish rather than Irish. So that might be Mathieson or Mathewson. Both common Scottish surnames to this day.
My feeling therefore is that your ancestors are probably descended from Scots who arrived in Ireland in the 1600s (100,000 Scots settled in Ireland then). Your particular family might be related to the Irish Burkes – I can’t say - but in general terms I suspect it’s unlikely. I think the Burkes were from Mayo which is a considerable distance from Antrim, and not somewhere significant numbers of Scots settled.
There are almost no church records in Antrim from the 1700s and without an exact parish it’s unlikely you will locate any records of your family in Ireland.
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 are running an Ulster DNA project and can offer FTDNA testing kits at a reduced price. http://www.nifhs.org (Go to DNA project on the website).
