Well, not in the way you describe it, no.
It's theorized (and very likely) that when a star the size of our sun eventually becomes a red giant, it will then compress itself into a white dwarf. White dwarf stars are made of carbon almost entirely, compressed so powerfully that they've become diamond.
Here's the thing, though. The gravitational pull of these white dwarf stars is so powerful that even a grain the size of a piece of gravel would weigh several tons on the surface. This means that no pieces would be flaking off.
Now, even though the answer is "no," there might be several ways to get around that, and still keep the visual in your story. As I mentioned, a white dwarf is EXTREMELY dense, which means it would have a very powerful pull on any planetary debris orbiting it. I say 'debris,' because when your star went red giant before its death, it incinerated all planets near enough to it, expanded its gravitational pull, and possibly changed the orbit of any of its outer planets.
Perhaps, when your star started to die, an earth-like planet, possibly larger, was attracted toward the star and broke up in the process. The star's rotation could cause the material from the destroyed planet to orbit around it.
When debris first starts orbiting, it isn't necessarily in a perfect ring. It's messy. Eventually, the debris will align itself both to the star's orbit and to its highest area of gravitational pull, which would be the widest part of the star. That's why our solar system is aligned the way it is (mostly!) and why the rings of the gas giants orbit how they do.
However! At first, it might look as if the dead planet's debris is actually part of the sun, floating aimlessly around it in large chunks.
A slightly easier (And slightly less believable - in my opinion) way would be to introduce the concept of a technology that could freeze a sun. If I remember right, I think it was Farscape that used the infant concept of flash-freezing the entire star. Insert science here to explain why it might appear, at that point, that pieces of the sun's crust were floating away from it.
The thing to keep in mind through all of this is that a star has an incredibly intense gravitational pull. That's one of the reasons it's a sun, and not a large gas giant. The gravity inside a star is so intense that it begins fusing hydrogen into helium. Even in its death, anything other than light would have a very hard time escaping that pull.
Edited to add: If something caused a planet to explode, the core might possibly be dense enough to burn IF there was enough oxygen in the planet and its escaping atmosphere to cause it to do so. This would burn out pretty quickly - relatively. You could possibly push a century and still hold believability. Your character might mistake this planetary core and the debris surrounding it as a dead sun.
