Jack family

You don't say what denomination your family were but sadly no denomination in Ardstraw parish has any records earlier than the 1830s so it may prove very difficult tracing this family.

Looking at the 1901 census, there are 82 people named Jack in it in Co. Tyrone. 6 were Church of Ireland and the remainder Presbyterian. Presbyterianism was brought to Ireland by the Scots and is a very strong indicator of Scottish origins. In addition, the surname Jack would not be regarded as a native Irish name. So that profile points very strongly to all of them being of Scottish origins. So the answer to your question, whether your ancestors came to Ireland in the Plantation is probably yes. Almost certainly.

There’s no list of individual families who came in the Plantation. The experts say that in the course of the 17th century some 100,000 Scots settled in Ireland (representing about 10% of the entire population of Scotland at the time).  If you want to read about that in more detail, I recommend: “The Scottish Migration to Ulster in the Reign of James I” by M. Perceval-Maxwell which covers the initial settlement of Tyrone (and the rest of Ulster) or "Eagles Wings - The journey of the Ulster Scots and Scotch-Irish" by Dr David Hume MBE which covers all of the 1600s.

The names of the big landowners are known. A condition of being granted lands in Ireland was that they brought a specified number of tenants with them from their Scottish Estates. The area around Ardstraw (the Barony of Strabane) was set aside exclusively for Scots, and was mainly settled by James Hamilton, Duke of Abercorn, who came from Renfrewshire (just outside Glasgow). So some settlers in Ardstraw will have come from that county. However I am not certain that the Jacks were amongst that first wave. One of the other conditions of being given land in Ireland was that each of the big landholders had to ensure that the settlers were armed and could defend themselves. As a consequence Muster Rolls were kept naming nearly all the adult males, and their weapons. Looking at the Muster Rolls: “Men and Arms, the Ulster Settlers c 1630” edited by R.J.Hunter, there are no Jacks listed in Co. Tyrone. There were 3 in Co. Derry though. So it’s possible that the Jacks arrived in a later wave of migration. (A lot of settlers came in the last decade of the 1600s due to famine in Scotland).

There’s no Jack households listed in the Hearth Money Rolls of 1666, again suggesting a later arrival:

http://cotyroneireland.com/hearth/ardstraw_1666.htm