No. They're one-shots -- that's why they're so highly valued.
Edit to answer question in comment: The best way to find these unlockable missions seems to be the slicing skill -- they show up as loot from bountiful/rich lockboxes sometimes.
It's actually sort of confusing. There is a lot of posts out there, including from official SWTOR sites that don't tell the whole story. Most of them suggest that there aren't inherit stats. Which is sort of true, but not totally:
From SWTOR Stats & Attributes:
Due to the extreme amount of gear customization in TOR, developers
want you to be able to take any orange item (orange = fully moddable)
and use it all the way to max level if you prefer. In order to achieve
this, weapons and armor both have a dynamic stat on them which changes
based on the level of the mods you have placed inside the item.
- Rating – Increases the damage done on a weapon. You’ll see this listed
at the very top of a lightsaber or blaster’s tooltip. The stat itself
does absolutely nothing except increase the damage done — think of it
as the ‘item level’ stat you’re used to seeing in games like World of
Warcraft.
- Armor – Armor, which you should be more familiar with,
increases Energy and Kinetic resistances (more on that later), is the
dynamic stat increased on pieces of equipped gear.
So, for an example, if you take an orange lightsaber with all level 1
mods in it, compared to the same lightsaber with all level 10 mods in
it, the Rating stat will be much higher, and thus so will it’s damage.
Same goes for Armor on those pieces with Armor on them.
(Which is a good summary from this really long SWTOR.com forum thread that did have an op post in it that clarified the situation to what we know now.)
The op explained that this dynamic stat is important so that modded items can drop from bosses on instance and still have some inherit value so that the first boss in an instance isn't farmed over and over for only the mod pieces.
The above SWTOR.com thread also discusses one important bug that is confusing a lot of people. If you do not equip the weapon/armor or relog after modifying it, the tooltip will not accurately reflect all the correct stats based on this dynamic stat modifier. Depending on what you change, when, and what is in it now, the numbers in the tooltip can be a lot higher or lower than they really are.
UPDATE:
Some additional clarification from the official devs:
Orange Armor gets it's armor value from the 'armoring' mod you slot
in. If you exchange it for something better, the armor value increases
along with the stats.
It's always the first mod in the list that determines the intrinsic
properties of the orange items (e.g. armoring on armor, barrel on
guns, etc.)
Best Answer
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.