There is no direct evidence outside the Bible that a nation of Hebrews were , as a nation, slaves in Egypt.
It is not just that we don't have Egyptian evidence, it is that the Exodus is followed by the Conquest which implies that the freed Israelites then moved into Canaan, as a distinct nation, and took it over with force. Archaeologists like William Dever deny we have evidence of this either (or rather, we have reason to doubt that the cities claimed to be destroyed by the Israelites were in fact destroyed by them). The Israelites look very similar to the Canaanites they were supposedly distinct from archaeologically for a long time, there is no need for a conquest model. They were mostly Canaanites who evolved a new self-identity.
The implications of the new picture of indigenous Late Bronze Age Canaanite origins for the majority of the early Israelite population is clear. Not only is there no archaeological evidence for an exodus, there is no need to posit such an event. We can account for Israelite origins, historically and archaeologically, without presuming any Egyptian background. As a Syro-Palestinian archaeologist, I regard the historicity of the Exodus as a dead issue.[1]
Egyptologist Jan Assmann is even more blunt:
Such questions about historical reality end in a blind alley. The biblical stories contradict each other, while extrabiblical sources are in short supply. The “historical Moses” has vanished into thin air and a verifiable exodus cannot be reconstructed from the stories.[2]
Bible scholar Joel Baden speaks on the textual side (he more or less agrees with the conclusion of archaeology reached by Dever)
There are two major categories of data to explore: material artifacts and textual records. Within the first, we might well expect to find some material remains either from Israel’s four hundred years in Egypt or from their forty years in the wilderness. These were, after all, humans, living human lives: building dwellings, consuming food, discarding waste, participating in communal celebrations, dying and being buried. We need think only of the various discoveries of Paleolithic remains in Africa, or Neanderthals in Europe, to recognize that even much smaller groups of people cannot avoid leaving some trace of themselves behind, be it only a single cave painting, a solitary jawbone, or a stray flint.
And yet, when it comes to the Israelites of the Exodus, either in Egypt or in the wilderness, we have nothing. Not a single material trace of Israel’s presence has been discovered. I remember mentioning this phenomenon at a family Seder one Passover, and my grandmother clinging to hope: “Perhaps they just haven’t found it yet.” It must be admitted that the lack of material remains is an argument from silence: absence of proof is not proof of absence. But neither the settled regions of Egypt nor the Sinai desert are so large that we should have simply failed to discover the places that the Israelites would have lived and traveled. Especially in this era of aerial photography and Google Earth, it is hard to imagine that the material remains of two million Israelites can have slipped under the radar.
On the textual front, unsurprisingly, the situation is similar. Egypt, like many advanced ancient cultures, kept quite good records: lists of their successive kings, annals of their political and military engagements, even registers of the annual level of the Nile. In all of ancient Egypt’s archival records, in all of its official correspondence, there is no mention either of the wholesale enslavement of the Israelite people or of their precipitous departure. There is not a word about the land being ravaged by a succession of plagues or of the death of Pharaoh and his entire army at the sea. It might be argued that such embarrassments were swept under the carpet in Egypt’s formal records, but it seems rather unlikely that the miraculous watery demise of a pharaoh could be easily glossed over.[3]
But Baden warns: "If the archaeological evidence suggests that the Exodus story did not take place as described in the Bible, this does not necessarily mean that there is no factual basis at all for the narrative"
This is not to say that there aren't attempts to find some historical kernel or cultural memory that acts as the basis for the story.Some people point to the Egyptian name of Moses, and the Egyptian names of the Levites (I believe Friedman uses this to argue that a small tribe of Levites were the real characters of the Exodus, not 600,00 men- to say nothing of families)
Some people point to the Hyksos, a Semitic people who ruled Egypt, were expelled (and allegedly destroyed), others to the apiru a free-moving class of mercenaries/vagabonds and so on. All of these have been challenged and debated.
Others point to the fact that Egypt ruled Canaan for centuries before its rule collapsed. I read an interesting article[4] about Egyptian ideology which essentially insisted that all of Canaan and Egypt's controlled territories were slaves of the Pharaoh. The Pharaoh would literally put images of defeated enemies and subject peoples on his floors or his sandals so he could tread on them as he walked. No doubt the Canaanites would have been glad to see the back of him. And, with the collapse of Egyptian control, Canaanites who had been slaves or corvee labor (drafted to build things for the Egyptians) either in Egypt or in Egyptian-controlled Canaan may have flowed back home, or developed a story of being rescued from Egypt the state (which eventually became Egypt the land)
In this light the Exodus would be a counter tradition attacking that Egyptian ideology. Possible the Israelites relocated the site of their "slavery" to Egypt to make them distinct from the local Canaanites as a method of identity formation; the Hebrew Bible is replete with examples of the problems that come with identifying too closely with such nations which inevitably leads to religious corruption, so this gives them some distance.
"In this spatial displacement, the Exodus from Egypt becomes the narrative site for forging a new people, who enter Canaan with clearly marked external ethnic boundaries. In this sense the spatial movement overcomes the problem of the basic lack of cultural difference between early Israel and its Canaanite neighbors. Through the strategic processes of cultural memory, Israel projected its origins to “outside” in order to construct a distinctive identity “inside.” As Peter Machinist emphasizes, “A story of outside entrance into Palestine . . . would have served as an important pole around which a collective identity could be segregated and consolidated"[4]
Anyways: we can't know. We cannot prove the negative that some small group of slaves -or even a continual trickle of slaves from Egypt which was the master of Canaan for centuries- couldn't have come back and acted as the kernel that later gave rise to the whole story. We cannot rule out that the legacy of Egyptian domination of Canaan (as opposed to domination of Israelites within Egypt) didn't form the cultural memory that gave rise to the story. But we cannot archaeological find them and the majority of Israelites are seen as coming from the region, not out of it.
Of course, even if we had the historical kernel, the story has grown into something much grander. That has to largely be considered myth or unverifiable.
References: