Hindu people believe that the Cosmos is dying and emerging in a never ending circle.
How is this seen by Jews and could I be born in the new Cosmos instance as a Jew, even if I am not?
Kabbala (Jewish mysticism) talks about there being multiple worlds, but our laws of who-is-a-Jew basically pertain to the world that we know right now.
Reincarnation is a concept stressed by kabbalists starting in the late 1500s, though some traditionalists challenged it. Today I'd say most rabbis have heard of the concept, but if someone doesn't believe in reincarnation they're certainly not a heretic. (Note that we do believe that there will be a complete resurrection of the dead at some point in the future.)
The kabbalistic explanation of a convert is in fact a Jewish soul that was born into a non-Jewish body, hence it goes searching for its roots. Kabbalists also describe one possible punishment for malicious slander as reincarnation as a dog.
I'm not aware of anyone discussing a non-Jewish soul reincarnated as a Jew (unless this version of the non-Jew was in fact originally a Jewish soul).
As my Talmud teacher liked to say, eh, could be ... could be not!
We are a lot more focused on what we do with the souls, bodies, lives and situations that we have right now.
Judaism does not view the universe as eternal (Genesis 1:1). Only the one and only God is eternal hence there is no question according to the perspective of Judaism.