I was hunting through the board for the same information as you, as I am also descended from Robert Hunter (b. ca. 1714 in Ireland?, d. 1783) who married Elizabeth Gillespie...My line is:
Matthew Hunter (1750-ca. 1804) m. Antje Crist (ca 1748-ca. 1802)
Elizabeth Hunter (1773-1844) m. Jacobus Smith (1769-1846)
Matthew Smith (1795-1876) m. Eliza Moule (1801-1839)
Harriet Smith (1828-1882) m. Nelson VanKeuren (1826-1907)
Henry Smith VanKeuren (1852-1927) m. Kittie May Palmer (1853-1917)
Leota VanKeuren (1896-1990) m. John Howard Stitt (1893-1962) (my grandparents)
At any rate, given the estimated birth date of Robert, I'm guessing his father, James Hunter, would have been born in County Armagh around 1670-1690.
Anyone with a suggestion where to look next?