Glad to help Mary. Assuming that I have transcribed the lyrics correctly, I would guess that "belonging to the right" refers to his having been on the 'right' side (not meaning 'right-wing') in some political issue of the day - but whether American or Irish politics is meant I can't tell.
I don't know how early the Irish became engaged in American politics, and yes I do know that the Scots Irish played a significant role in the War of Independence, but I am thinking more of the generation who migrated in the 1840's and 50's.