Just to be clear: no Jews became slaves in Egypt, just as no Canadians died in the Black Plague of the 14th century.
The ancestors of the Jews, who were known as “Israelites” (since they were all descendants of Jacob, whose honorific was “Israel”) were enslaved, according to the Book of Exodus.
The concept of “Jewish” (or “Judean”, to be exact, if we’re talking of biblical times) emerged only centuries later, following the settlement of the Israelite tribes in the Holy Land, in relation to the descendants of Judah (one of Jacob’s twelve sons)—more specifically, to the southern kingdom, following the split of the Israelite kingdom after King Solomon into two: “Israel” in the north, and “Judah” in the south.
When the Israel kingdom was conquered around 725 BCE by the Assyrians, its population was scattered (forcibly relocated to other parts of the Assyrian Empire, in keeping with Assyrian custom, to minimise the risk of insurgency), and some managed to find refuge in the southern kingdom, where they and their religious traditions were absorbed into the Judean nation, at the instigation of King Hezekiah.
Thus, the Judeans became the sole preservers of the original Israelite identity and tradition. As it happens, the Exodus tradition was one of those northern (Israel) traditions—which suggests that it was the northerners’ ancestors who had been in Egypt, not those of the Judeans, and that the story of the twelve brothers/tribes was conjured up during the integration of the Israelites into Judea as part of the effort to forge a joint mythology, to make it look as though they had been a single nation from the start.
This explains, among other things, why we see no connection in the Bible between Jacob (the ancestor of the Israelites) and his putative grandfather, Abraham (the ancestor of the Judeans)—and why Jacob’s ancestral homeland, which he is sent to by his father Isaac, is Haran (southeastern Turkey of today), while Abraham’s is in Ur of the Chaldees, in modern-day southern Iraq.
For more on the synthesis of Israel and Judah, see Shamai Gelander’s book.