Exactly why would the Yronwoods fight for the Blackfyres when they were anti-Dornish and pro- war? Did they think that Daemon would let Dorne go off as an independent nation under the Yronwoods?

Because not all of the Dornish agree on what Dorne is or ought to be. From the Yronwoods perspective, the Martells and  all the Rhoynish are colonialist bastards who took their land. 

So I think the Yronwoods fought for Daemon Blackfyre, because Daemon’s pro-war solution to the Dornish question was to crush the Martells, rewrite Daeron’s treaty, and make House Yronwood the new Lords Paramount of Dorne.