I would like to know espescially from a kabbalistic viewpoint: Is it possible that a Jew gets born into the body of a non-Jew because of reasons connected to previous lifetimes and has to find his way back to Judaism as a sort of purification process?
The Ohr Hachaim in Parshas Ki Tetzei says in regards to the the halacha of Eshes Yefas Toar-The captive woman at the battlefield. that the soul of this non jewish woman really comes from the spark of a holiness that wants to cleave to HaShem.
The Gemara says (Avodah Zara 2b) that HaShem went to all the nations and offered the Torah, and the nations rejected the offer, however there were Gentiles who did want to receive the Torah, but they were overruled by the majority, thus we say that Gentiles who choose to convert are really those gentiles from the past that really took upon themselves to keep the Torah.
Clearly, the implication of theses sources is that a gentile cannot have a Jewish soul, but instead may have a gentle soul which is drawn to Judaism