I tend to play my Beth games as a Lone Wanderer, but when Companions have unique quests then I like taking them around until those quests are done.

So my question is: do the companions have their own quests? Or can I leave them alone without regrets?

Some have there own quest, you'll get them when they like you enough but not all companions have quest.

Yes they have their own missions. Not every companion has a mission since Dogmeat doesn't have a mission. I believe companion missions are acquired at a certain relationship level and since Dogmeat already has maximum relationship when you get him, then there is no mission to unlock.