A few things to consider, which may echo material other posters have contributed but it all bears consideration.
First, the Hebrews were an entirely separate ethnic group from the Egyptians. The languages, traditions, religions, and material culture of the two groups are distinct and recognizable. The origin of the ancient Egyptians represent a genetic blend of North Africans, as cormac mentioned in an earlier post. To that I would add prehistoric populations from the southern Levant, who settled largely in Lower Egypt. Egypt solidified into a state in around 3100 BCE, and developed mostly independently and without foreign incursions for the next 1,400 years. This is why the pharaonic Egyptian culture is so distinct in flavor, and is why its traditions remained so fixed for more than 3,000 years.
The exact origins of the Hebrews are not as clear, and the debate persists. However, nearly all modern scholars agree that the Hebrews originated in Canaan and Transjordan. This happened only at the end of the Bronze Age (c. 1200 BCE) but centuries passed before a cohesive, recognizable, cultural entity emerged that we could call Israel. Some element of Israelite people seems to have existed in the late thirteenth century BCE, based on the victory stela of Merneptah (c. 1207 BCE), but what "Israel" meant at that time is not clear. Going by how the Egyptians described the earliest Hebrews, they appear to have been a loose grouping of nomadic or semi-nomadic people. Whatever the case may be, the victory stela seems to point toward the very earliest beginnings of a people who were seeing themselves as something different from their Canaanite kin.
Prominent scholars like Israel Finkelstein and Amihai Mazar might disagree on specific points but share a consensus that the Hebrews did not come from a foreign land. This is the gist of the point I'm making: the Israelites emerged as a culture right there in Canaan. Definitely not Egypt. William Dever has argued that the Hebrews emerged from coastal Levantine city-states during the turbulent period of collapse at the end of the Bronze Age, and I find a lot of credibility in his position. Finkelstein has demonstrated through his own exhaustive archaeological excavations and analyses that there were previous movements of people in the Levant going back to the Early Bronze Age (c. 3000 BCE), during which populations settled in the lowlands and highlands of the Levant and fluctuated between pastoralism and urbanism; the Hebrews were just one more example of this.
What's clear in the archaeology of the Holy Land is that the material culture of nearly all periods and peoples of the Levant points to shared traditions and cultures. The Hebrews were no different. Although they quickly developed distinctive cultural practices like the proscription against consuming pork, the way they lived, the tools they used, the ceramics they produced, and other hallmarks show little difference from their Canaanite neighbors. Even the old identification of the four-chambered house as being distinctly Hebraic has been laid aside, as similar architecture has been excavated from purely Canaanite contexts.
I strongly agree with questionmark that the earliest Hebrews were not a homogenous group. They were more than likely a mix of people, including Canaanites, Sashu, Syrians, and other Western and Northern Semites. However it happened that these people banded together to create a new culture that became noticeably different from the others, it happened only because of the collapse of the key powers at the end of the Bronze Age. The power vacuum left behind by the fall of the Hittites and the weakening of the Babylonians and Egyptians, allowed new territorial states to develop in the Early Iron Age.
Sorry to drone on to such a degree, but this is a debate I've always enjoyed: the emergence of the Hebrews. On a note more specific to some things that have been mentioned in this discussion, the Hyksos and the Hebrews definitely were
not the same group. I can't stress that enough. While the Hyksos were predominantly Canaanites, this does not imply or establish a connection. The Hyksos were strongly polytheistic, as is attested by their shrines, temples, and burials. Moreover, they came onto the scene and disappeared from the scene a very long time before the first Hebrews emerged.
Several years back I started a discussion on the mistaken notion that the Hyksos and Hebrews were the same people; the discussion also includes the same misconceptions about the Habiru. For those interested, the discussion can be reviewed
here.
The word "Amen" cannot be used as a line of evidence, either. I've seen posters mention this for many years now, and in numerous different forums to which I've belonged. In the Hebrew language "Amen" is a declaration used in prayer. In ancient Egypt it was the name of an important god. However, it must be stressed that the spelling "Amen" for the Egyptian god is an artificial construct drawn from the skeletal consonants used in the hieroglyphs. We are not certain what the vowel sounds may have been for this god, which is why in Western languages today the name is also spelled as Amon and Amun. I personally favor the last, for no reason I can explain. The name is transliterated as
aimn or
imn, but that's as much of the pronunciation as we have. In ancient Egyptian the name meant "Hidden One."
Akhenaten has been mentioned in this thread. Some posters have done a good job dismissing any connections between his religion of Atenism and the later Hebrew religion, but I will stress that no evidence for Hebrews is extant for the period of Akhenaten (Dynasty 18, fourteenth century BCE). The so-called Amarna Letters, which date to the time of Amnuhotep III and Akhenaten, reveal no hint of a Hebrew-like people in the Levant. And perhaps most importantly, large-scale proscriptions against Akhenaten and his form of religion began almost immediately after his death and were vigorously pursued in the early Ramesside period of Dynasty 19. In all likelihood, by the time the Hebrews were becoming a recognizable culture, they would've had no idea who Akhenaten was. It's safe to say few Egyptians would've, for that matter.
All told, the Hebrews emerged in the Early Iron Age from the Levant itself. There is little reason to doubt this. While ancient Egypt no doubt had some influences on the early Hebrews and how their state and culture developed, the Hebrews were neither Egyptians nor, to be certain, a carbon copy of pharaonic Egypt.