Laura,
The problem you face is that there are no church or other comprehensive records of Rashee in the 1700s.
I searched in Griffiths Valuation (1862) for McQuitty in Rashee parish. There is only one household listed. That?s for a John McQuitty on plot 7, which was a 4 acre farm. He lived in the townland of Tildarg (on the modern B94 Collin Rd). He left that farm between 1863 and 1866. I looked in the 1901 census and there were no McQuitty households in Tildarg then. The nearest were these:
http://www.census.nationalarchives.ie/pages/1901/Antrim/Ballyclare/Ballyclare_Town/917980/
http://www.census.nationalarchives.ie/pages/1901/Antrim/Ballyclare/Le_Ballyclare/918574/
You don?t say what denomination your family were but the ones in the 1901 census were Presbyterian and Reformed Presbyterian (both obviously indicating Scottish ancestry). So I?ll guess that your were of one of those denominations too. There are several Presbyterian churches in the area. For example Ballyeaston 1st which has records from 1814, Ballyeaston 2nd with records from 1821, Ballyclare (1857), and Ballyclare Non Subscribing (1837). The nearest Reformed Presbyterian church is Ballyclare. Its records don?t start till 1867.
So none of those churches has any records that would cover William McQuitty?s marriage or births of his children.
I found these probate abstracts for McQuitty households in the area:
McQuitty David of Ballypallady House Ballypallady Doagh county Antrim retired farmer died 1 September 1956 Probate Belfast 16 October to Agnes Thoburn Gourley spinster and Robert Cyril Alexander works manager. Effects ?537
Probate of the Will of Catherine M'Quitty late of Ballee County Antrim Widow who died 1 January 1897 granted at Belfast to Catherine M'Quitty Margaret J. M'Quitty and Henrietta M'Quitty all of Ballee Spinsters.
Probate of the Will of James McQuitty late of Ballypallady County Antrim Farmer who died 2 May 1911 granted at Belfast to Samuel McQuitty Grocer and David McQuitty Farmer.
McQuitty Robert of Bruslee Ballyclare county Antrim farmer died 28 November 1935 Probate Belfast 27 January to Catherine McQuitty the widow and Samuel Logan farmer. Effects ?265 2s.
McQuitty James of Craigolive Ballyeaston Road Ballyclare county Antrim textile clerk died 21 January 1956 at Montgomery House Purdysburn county Down Probate Belfast 4 May to Samuel Agnew farmer and Robert Harold Galbraith presbyterian minister. Effects ?30.
I can?t say that any of the above households are your family, merely that they lived in the Rashee area.
If you are coming to Belfast or Rashee at the weekend, I am happy to meet you if you want. I can show you where Tildarg is, and the general area around Rashee. If so, I?ll send you my contact details via a PM.
Elwyn