It depends on where they were born in County Down and County Antrim. The map link I gave you was the Poor Law Commissioner's report and was based on those taking rashons through the Poor Houses (Workhouses) and not the general population. There's a list of the names of all the towns and townlands in Counties Down and Antrim at http://www.seanruad.com
and pick either county from the drop down menu.
The links I included in my previous message included book lists too. You can probably get many of them through interlibrary loan.
Since civil registration of births didn't begin in Ireland until 1864 church parish records for their parishes in County Antrim and County Down would be the sources for baptismal records for them. The Public Record Office of Northern Ireland (PRONI) in Belfast, County Antrim has many Presbyerian church records on microfilm, see: http://www.proni.gov.uk/
To find out where they were born in County Down or County Antrim you can search the following records in the US:
Irish born: civil and church marriage and death records, cemetery records, tombstones, wills and obituaries as well as US naturalisation records.
Their children's civil and church birth/baptism, marriage and death records, obituaries may include their parents birthplaces in Counties Down and Antrim.