What happens with a companion's quest if I dismiss one in Fallout: New Vegas?

Can I hire the companion again and continue the quest?
Do I need to start the quest from beginning?
Regarding Boon, does it count if I fight the Legion without him and then hire him again?

Boone's quest (Also Raul's, Arcade's & Veronica's) is triggered by earning "history points" by doing certain things while he is with you. You won't lose progress if you dismiss Boone, nor will you lose any quests from your log, but you can only make progress while he is with you.

This can make Boone's quest impossible to complete as many of the history points involve quests or killing certain NPCs and are not repeatable.

Note that this only applies to dismissing Boone through dialog. If Boone leaves or turns hostile because of your reputation or actions, you will fail any of his quests.

It's remarkably hard to find information on how Fallout:New Vegas deals with companions and their quests in this respect. Based on my own experience, though, and as I cannot find proof to the contrary, dismissing companions should not have any bearing on their companion quests.
From here:

When you dismiss a companion they'll return to their starting location and you can ask them back later. I'm not aware of any problems concerning the companion quests.

Something important to take into consideration concerning Boone:

if you want to do Boone's quest, don't do a lot of the anti-Legion side quests, or you might miss out on all the 'history' points you have to gain to be able to trigger his quests. eg, don't do tons of those things without him.