DianaN,
The guide to Irish Quakers that I read said they were pretty well unknown in the west of Ireland, primarily because it didn’t have the commercial opportunities they sought. There was little or no industry there then. (Looking at the 1901 census of Co. Clare, there isn’t a single Quaker in the county). There are no records of any Meeting Houses in Clare, ever. There just weren’t any Quakers there. If the Neal family originate in Clare it’s highly unlikely that they were Quakers then.
The first Quaker meetings in Ireland were around 1654 in Lurgan, Co. Armagh, when the denomination first arrived from England, so any timescale for the family becoming Quaker really has to be after that. They probably moved from Clare to Dublin or Ulster for work or commercial reasons. Famine wasn’t a significant issue in the 1700s (the 19th century population explosion, combined with blight caused the famine in Ireland). Some other factor likely caused them to move if they were in Antrim in the 1700s.
https://en.wikipedia.org/wiki/Quakers_in_Ireland
Moving up in Society is an interesting suggestion, but if that was their motive, why not just become Church of Ireland? Friends have a fairly strict code of conduct and weren’t slow to reprimand or expel those who didn’t abide by it. The Monthly meetings are full of investigations into Friends who were alleged to have drunk alcohol or conducted their businesses in an unprofessional way. And of course they couldn’t marry out. They had to marry another Quaker, or be expelled. That caused a lot of difficulties. They resisted the tithes and so were frequently in trouble for refusing to pay them. A tough life in some respects. Not everyone’s choice. And in spite of perceived English hierarchies, there were many wealthy Catholic merchants and business people. You didn’t have to leave that denomination to improve yourself. There was a Catholic upper class in Ireland, and a middle class developed all through the 1800s. Asenath Nicholson was a wealthy American who toured Ireland for a year in 1846 visiting every county but one (Cavan). Her account of her travels was published as “Ireland’s welcome to the stranger.” Well worth reading. She frequently mentions meetings with wealthy and successful Catholic families. She noted that many Upper Class native Irish looked down on the poor as disdainfully as their English counterparts. (She also commented that not all the good landlords were Irish and not all the bad ones were English).
I suspect your ancestors must have become Quakers because they particularly identified with Quaker values. But I have no way of suggesting how you might ask them if that’s correct!
Very few Irish Catholics migrated to North America in the 1700s. The vast majority in that century were Protestants, and those were mainly Presbyterians (ie descendants of Scots who had settled there in the 1600s). Some Church of Ireland and other denomination did go too but the majority were mainly from Ulster and were mainly Presbyterian. Roman Catholics (ie native Irish) started to migrate in big numbers in the 1800s. The migrants who left in the 1700s were mostly descendants of people who had already made one move so were more accepting of the idea of moving again, but in addition they tended to be slightly better off financially and so able to afford the journey. They left for some of the same reasons that led to Catholic migration in the 1800s ie lack of land, unfair rents, inability to buy land, poor harvests, low prices for cotton and linen, religious restrictions imposed by the state, and so on. Just they were more inclined to go and, in the case of Presbyterians, their Ministers often encouraged them to go. They often accompanied their congregations to North America. So perhaps a bit better organized too.
