What position does Thoreau take on the conflict between the majority rule and individual conscience?

You've practically answered this question by asking it. Thoreau believed - and lived up to his beliefs - that just because MOST people were able to agree on a rule, that didn't necessarily make it a good rule.

This was during a time when the notions of majority rule and "Law and Order" had reached almost religious veneration. Today we give a little lip service to the notion of "minority rights," but in many ways we continue to act in accordance with Roman dictum, "Vox populi, vox Dei" (The voice of the People is the voice of God) which often leads to the "tyranny of the majority."

Some disturbing experiments in the 1950's showed the people were willing to do all sorts of horrible things as long as they were sanctioned in these by an authority figure; as long as they weren't, in the Words of so many authority figures, "taking the law into their own hands." Thoreau's point was that the individual conscience was still in charge of each person's behavior, no matter what the "law" said.