No, Adam and Eve were not Jewish.
The word “Jew” is a contraction of the Hebrew word “Y’hudah”, Judah. Jacob had twelve sons and his sons became the fathers of the twelve tribes of Israel. When Jacob’s wife Leah gave birth to their fourth son, she praised God and celebrated the birth. The name Judah means celebrated or praised.
God specially honored Judah with this promise: “The scepter will not depart from Judah, nor the ruler’s staff from between his feet, until he comes to whom it belongs; the nations will obey him.” Genesis 49:10 (NET)
The scepter and ruler’s staff refer to kingly authority and power. Who will reign as king and receive the obedience of nations? Only Jesus. Thus, this verse means Messiah will be born from the tribe of Judah. Jesus’ genealogy is traced back to Judah in Luke 3:33, “the son of Judah.”
Today, people use the noun Jew to refer to anyone who professes to live under the Law Covenant given to Moses at Mt. Sinai or who is a descendant of Abraham through Jacob.
Gentile is a very general term. Today, people use gentile as referring to everyone who is not Jewish.
