It doesn't matter how long people on the individual pieces can survive, they will be killed when the pieces are sucked into the gravity well of the sun/star.
This isn't a planet that cracks open, yet the pieces still hang around each other because there's not really any other place to go. It's not even a moon that has a stable orbit and the pieces don't have enough change in direction to impact the plant they orbit. This is something that can't be roughly assumed to be a point mass, because it acts completely different. It would act more like a rubber band around a ball being cut, rather than the rubber ball having a section cut from it. But even the rubber band around a ball isn't a good analogy.
A Dyson ring may have some angular momentum if it was rotating around the star, due to it needing some way to help maintain a stable position. With this motion, some pieces may be flung away from the star and some pieces may be flung directly into it (or near enough) so that it melts and is destroyed.
A Dyson ring may not need to be rotating around the sun, so now you have pieces of debris that are relatively stationary to the star.
Even if pieces were flung away from the star, they will still be drawn back into the star because it's a huge mass and pulls things into it. The only reason planets, comets, and other natural debris isn't pulled into it is because they are moving at the correct speed and angle to avoid the star. Going the wrong speed or at the wrong angle, and the star adds more mass to itself.
The likelihood of a piece being ejected from a destroyed Dyson ring at the correct speed and angle to achieve orbit is nil. The likelihood of a piece being ejected from a destroyed Dyson ring at the correct speed and angle to achieve escape velocity is even less.
The best you can do is to send a rescue to each section/piece in order of how desperate and how likely they are to die soon. So this isn't just about individual sections supporting life, it's also about triage of who can survive to be rescued and what order they need to be rescued in.
Some of these pieces may not be pulled into the star for days, weeks, months, or possibly years, so yes, you still need to decide if those pieces remain habitable and for how long. But they will all be destroyed by the atomic fires of the star they once surrounded.
I realize this significantly changes your story, because now you have to decide if there's a civilization that wants to do the rescue, if this civilization has the ability to perform it's own rescue, if the interstellar community watches this evil civilization disappear with a sigh of "good riddance", or some other outcome.
I didn't explicitly say it before, but I'm making the assumption that simply getting near the star is enough for the debris to burn up to a point on inhabitability. I will admit that I didn't realize that the "no-go" volume around a star was so large, but it kind of makes this answer a little more on point.
Current human technology lets us get within about 3.83 million miles to the surface of our sun before it gets too hot. I'm assuming that a civilization capable of creating a Dyson ring isn't making one ~12 million miles in circumference. That said, whatever material they are making the skin of the ring out of, I'm assuming they aren't making the interior in that same material, so any broken ends not protected by bulkhead doors will be incinerated.
Also, only a fairly sizeable section (FSS) of ring debris could be assumed to have enough self-sustainment equipment to maintain a livable atmosphere in the extreme conditions around a star. (What "fairly sizeable section" means is definitely subject to interpretation.)
One issue with these FSS is that they will have enough mass that maintaining the current orbit or finding a different stable orbit will be next to impossible being that close to the star. I'd have to assume that maintaining orbit of even a moving ring would be considerably different than maintaining orbit of an individual piece. So it would also have to be assumed that this FSS would also have maneuvering thruster(s) to even have a chance of not being pulled into the star. Not only that, but the thrusters would have to be in the correct position and orientation to be able to put thrust in the correct direction to push the FSS into a stable orbit. And now we finally have to worry if there's enough reaction mass for the thrusters to even matter. Given all that, we still have to worry if the thrusters are capable of producing enough thrust, producing that thrust long enough while not going beyond expected usage, and if the FSS is structurally sound enough to survive the firing procedures.
Another issue is that we can safely assume that even these FSS aren't 100% self-sufficient in that they will eventually have to bring in more air, food, repair parts, etc. Granted, they may be able to harvest some of this from the plasma around the star and use the energy from the star to extend their lifetime, but they will still likely run out of something they need, eventually. So, if the FSS doesn't also include some sort of ship, manufacturing center, facilities to harvest plasma, and more, survivors of the initial disaster will not be able to survive indefinitely. Granted, this isn't part of the issue of falling into the star, but it's still a significant issue of survivability.