Your problem is that the births and marriage were all before the start of any records for that area. Ballynure Presbyterian church has baptism and marriage records from 1819 onwards. Too late for you obviously. If there were any earlier records they have long been lost. (Not uncommon in Ireland).
Gravestone inscriptions from Ballynure Old Graveyard:
This tablet to mark the burying place of the Ellis family of Straid & Prospect was placed here by direction of their last representative Jane Anne Clements Nicoley in Jul 1884. Tablet in opposite pillar has been placed there for safety.
Clements Erected by Mary McCurdy in memory of her husband Henry Clements who died on 02 Mar 1831 aged 75 years. The above-named Henry Clements' first wife Martha Simm died 09 Jul 1800 aged 48 years.
Clements Here lyeth the body of Frances Clements aged 82 years. 1812.
Clements In memory of Mary Clements, the beloved wife of Wm. Clements, who departed this life on 05 Mar 1867 aged 63 years. And an infant son. Also her son Andrew who was lost at sea in May 1852 aged 22 years. Also the above-named Wm. Clements who died 16 Mar 1882 aged 76 years.
Clements See Philips
Kirk Erected by Andrew and James Kirk of Ballynure as a small tribute of regard to the memory of an affectionate mother who departed this life 31 Jul 1824 aged 50 years. Also of their sister Ann Kirk who died 09 Jan 1809 aged 11 years. Likewise to the memory of their father William Kirk who died on 28 Dec 1841 aged 69 years. The remains of the above named James Kirk also lies here. He died 23 Oct 1844 aged 42 years.
Kirk Erected by John Kirk of Legaloy to the memory of his beloved wife Elizabeth Kirk who departed this life 02 Jan 1853 aged 62 years. Also the erector, John Kirk, who departed this life 26 Sep 1857 aged 77 years.
Kirk Erected by William Kirk of Clements Hill in memory of his son John who died on 26 Apr 1854 aged 27 years. Also his son James who died in Mobile, U.S. 1853 aged 23 years. Also their father William Kirk who died 14 Jul 1858 aged 67 yrs. Also his wife Elizabeth Kirk who died 03 Nov 1868 aged 70 yrs.
Kirk Erected by Robert Hughes, Larne, in memory of his wife Mary Hughes who died 16 Jul 1899 aged 70 years. Also her mother Agnes Kirk who died 20 Nov 1864 aged 80 years.
Kirk Erected by Jane Kirk of Hydepark in affectionate remembrance of her husband Thomas Kirk who died 15 Dec 1870 aged 45 years. Their daughter Sarah died 20 Jun 1865 aged 17 years. Their son William died 10 Aug 1865 aged 19 years. Also their infant children: John, Robert, and Mary Jane.
Kirk Erected in memory of Andrew Kirk, Marsden Villa, Belfast, who died 11 May 1875 aged 75 years. Also of Sarah, wife of the above Andrew Kirk, who died 05 Jul 1889 aged 65 years. J. Robinson & Son, Belfast.
Kirk Erected by Samuel & Jane Kirk of Clementshill in memory of their children: William Kirk died 24 Apr 1882 aged 15 years. Elizabeth Park Kirk died 02 Aug 1884 aged 11 years. Jane Kirk died 24 Aug 1885 aged 15 years. James Kirk died Jun 1892 aged 27 years.
Parents, for us let fall no tears; We early loved God's blessed Son, He took us home, and hushed our fears, And said, our earthly race is run.
Also their son David John died 10 Oct 1907. The above Jane Kirk died 16 Feb 1916. The above Samuel Kirk died 24 Apr 1916.
Kirk Erected to the memory of Alexander Kirk, Legaloy, who died 25 Feb 1896 aged 66 years. Also his wife Margaret A. Kirk who died 07 Jan 1900 aged 66 years. And their son James Kirk who died 08 May 1889 aged 24 years. And their son Thomas Kirk who died 01 May 1933 aged 70 years. Also Elizabeth, wife of above Thomas Kirk, died 11 Apr 1955. I know that my Redeemer liveth.
Kirk See Lorimer and McGookin
Source: http://www.ulsterancestry.com/newsletter-content.php?id=489
Ballynure was certainly heavily involved in the 1798 uprising. In general, the names of the rank and file supporters who fled after the defeat at the battle of Antrim have not been recorded. (No legal action was ever taken against them). A handful of senior commanders were tried and executed (McCracken etc) “pour decourager les autres” and some middle ranking officers were held and given the option of a trial for treason (penalty death) or being allowed to leave Ireland with all their family for good, usually within 3 months. A pragmatic way of ensuring they didn’t participate in any future rebellions. Some of their names appear in the Black Book of the Rebellion (MIC575/1 in PRONI, the public record office in Belfast). But if your family remained in Ireland till around 1817, I’d say they were rank and file supporters and probably don’t appear in any official records of the rebellion.