Are there any quests with more than one companion?

Since Patch 4.0, the below is less relevant — companion gear is now purely for looks, and their stats come from level, Influence, and Presence. So you never need to keep gear around for them.

There are a handful of quests (usually late in the story) where you must swap between multiple companions (though they are not present at the same time). The most notable is the Trooper, where I believe you make use of all of them. If you keep a set of your own hand-me-downs you could simply swap that set from companion to companion (apart from M1-4X) since they all use Aim gear as well.

Another example is where you must use T7 during one of the Knight's critical quests, but fortunately there is a broken droid right before the boss that contains droid gear. As far as I know, there is no quest where you have no recourse if you haven't kept your extra companions geared; I usually only keep one healer or DPS companion geared, and have played all of the stories to completion without issue.