I'm afraid I don't see any scenario where Northern Ireland could be peacefully unified with the Republic between the 1960s and today, nor in the foreseeable future. In the earlier years northern Protestants would be unwilling to give up their position of privilege and power, and would be absolutely horrified by the amount of influence the Catholic church had in Dublin at the time. And as time goes on a United Ireland becomes a much less attractive proposition to the Republic. Any reunification would be accompanied by levels of violence with which the Gardaí would be unable to cope and financial costs which even the strongest ebb of the Celtic Tiger would struggle to manage (Northern Ireland's oversized public sector, love of the NHS and inability to even come close to paying for itself would likely send the poor sod stuck as Minister for Finance at the time to an early grave) - not to mention a great deal of political and constitutional wrangling (you'd be stuck sorting out everything from electoral systems and the "special position" of the Catholic Church, to treatment of northern culture and the almost inevitable calls for some sort of devolved administration in the north).
Still, setting aside the question of how unification is achieved I'll return to the OP. Simply put, the Protestants of the north are very unlikely to peacefully integrate with the Republic at all. You'll have Loyalist terrorist attacks all over the place, protest marches, Ian Paisley et. al. roaring from the pulpit about the evil Papists and more nasty sectarian violence than the Guards and the Irish army will be prepared to handle alone (remember that for much of the Troubles the British army presence in Northern Ireland was more numerous than entire Irish army and police service combined). Even if things stay relatively peaceful there'll be loud calls for Northern Ireland to either rejoin the UK or go independent, either in its existing form or as a repartitioned entity (which would likely go from "majority Protestant" to "overwhelmingly Protestant" by getting rid of troublesome Catholic areas such as Newry, Derry and Fermanagh).