Adam and Eve were definitely not Jewish. Their wasn't Jewish yet. The question could start with Avraham and Sarah and about their status if they were "Jewish" yet or not.
Adam and Eve were not Jewish. But they were in a different category than what we know as Gentiles:
a) They were created as immortal;
b) They did not have a Gentile mother;
c) They were the ancestors of Noah, not the children of Noah.
Regarding Avraham and Sarah, please see the following answer:
"Avrahom, Sarah, Rivka (Rebecca), Rachel and Leah, and Bilha and Zilpa (who were half-sisters of Rachel and Leah), all belonged to the same extended family, and they were all spiritually Jewish as a result of the souls they were individually granted by G-d. But since the Covenant at Mount Sinai had not yet taken place, there was not yet a Jewish People, and they were still bound by Noahide Law.
These six women had the power to pass on their spiritual Jewishness to their children, through gestation in the womb, but Avrahom as the father did not, so Ishmael was not spiritually Jewish. Thus Sarah's only child, Yitzchak, was spiritually Jewish, as were Rivka's two children Ya'akov (Israel) and Esav (Esau).
All of Ya'akov's children (the Israelite family) from Rachel, Leah, Bilha and Zilpa were spiritually Jewish. Esav married out of this special group, and did not have any spiritually Jewish children.
All of these persons mentioned were granted souls of great spiritual, and thus historical, significance.
At Mount Sinai, the approximately 3 million assembled Israelites entered into a Divine Covenant with G-d and became the Jewish People, and they received the Torah Law from G-d through Moses. From that point on, anyone born to a Jewish mother is a Jew, and is obligated in the Jewish commandments of the Torah. Likewise anyone born to a Gentile mother is a Gentile (one of the Children of Noah), and is obligated in the Noahide commandments of the Torah."
https://asknoah.org/forum/showthread.php?tid=37