Short answer: No!
Long answer: I proposed that the "purpose" of the Stareaters was to consume stars before they go nova, supernova or hypernova. Our sun, Sol, does not have enough mass to go in that direction.
Instead, it will slowly consume all its fuel and become a red giant. We have about 5 billion years left before this happens.
So the Stareaters coming to take our star because it was going to go nova some day was a little bit of dramatic license. They did say they were curious as to why there was gravitic modulation emanating from our Solar System which was consistent with sentient life. Rome, through Aason, contacted them and dissuaded them from eating our sun. At least for the next million years or so.
So you will have to forgive me but it sure was fun to add in that second climax.