Egypt is the state with the most opportunity to have written about early Israel, and the one whose records are most often argued over by people who have not read what they contain.
What survives, in outline
Egyptian records bearing on Canaan fall into a few groups: campaign accounts on temple walls, topographical lists of places claimed as conquered, administrative correspondence, and triumphal stelae.
They are abundant for the New Kingdom, the period that matters most here, and they were compiled by a state with its own chronology — anchored by king lists and astronomical records — which owes nothing to biblical dating.
The Merneptah stele, and the weight it is asked to carry
One line on one granite slab, from about 1208 BCE, lists defeated Canaanite entities and includes Israel among them. It is the earliest mention of Israel anywhere outside the Bible, and it remains the only external fixed point near the period.
It establishes that the name existed and that Egypt knew it. It does not mention an exodus, a wilderness, a lawgiver or a conquest, and it does not say where this Israel came from. Both the argument that it proves the exodus and the argument that it disproves the conquest are asking one line to answer questions it does not address.
The campaign of Shoshenq, and the one place the records touch
A pharaoh the Bible calls Shishak campaigned in Palestine and left a list of captured places on a temple wall at Karnak. 1 Kings 14 records a campaign against Jerusalem in the fifth year of Rehoboam and the stripping of the temple treasures.
The Egyptian list is a genuine external record of an Egyptian campaign in the right region at roughly the right time. What it does not do is name Jerusalem in a way that settles the matter, and the reading of individual entries in the list is technical and disputed. This is a case where the external record and the biblical account are compatible and the fit cannot be tightened as far as either side would like.
The silence about the exodus
Egyptian records do not mention an exodus. Two conclusions are commonly drawn and neither follows.
The first is that the silence disproves it. But Egyptian royal records do not narrate reverses of any kind, and an administrative papyrus recording the departure of a labour force would be exactly the sort of ephemeral document that does not survive from the Delta, where the climate destroys papyrus.
The second is that the silence is therefore meaningless. That is too generous. Absence of evidence carries some weight when the evidence would be expected to survive — and for large events with monumental consequences, some traces would be. The honest position is that the silence is weak evidence, pointing slightly in one direction, and that it cannot bear the weight of a conclusion on its own.
This site grades that situation as unconfirmed rather than contradicted, and says why on the page rather than leaving the reader to infer it.
Related: the dossier on the earliest mention of Israel and the overviews of Exodus and 1 Kings.