Citation?
Well, likely the early Isrealites were “Apiru/Hapiru/habiru” but also just as likely the “apiru” didn’t nessesarily mean hebrew or isrealite. It seems like it was a generic rather pejoritive term to denote any of general nomadic attackers of the Egyptians from/in the general area of Isreal/palestine, etc… So, yes, the Egyptians would have referred to the early Isrealite raiders as “hapiru” but that term was likely not exclusive to the “12 tribes”. There are several passages in “The Oxford History of the Biblical World” that explain this, and the recent findings and interpretation of that word. Yes, at one time- what The Asbestos Mango posted was the prevailing mainstream thought. No longer.
The fact is- we don’t know what happened during that period. There is the “Conquest Hypothesis”, the “Pastoral Nomad Hypothesis”, the “Peasants revolt Hypothesis”, and a “Ruralization Hypothesis”. We do know that at the begining of the period, the early Isrealites weren’t in power- well… they didn’t hold any important cities anyway. At the end- they held pretty well all the cities. In between- those cities were conquered and re-taken, razed & burnt, (or fell by natural disaster) and rebuilt. Perhaps all of the “Hypothesis” have some kernal of the truth. Fact is- we don’t know, the evidence is too jumbled. Since the Isrealites didn’t start with any cities, and ended with them all, and nigh all of them have evidence of being taken by force sometime during that period- I think that if we assume that there was some conquering going on by the pre-monarchial Isrealites we are on safe ground. If you want to think that one of the great Generals of that time was named Joshua- that is certainly not unlikely at all- in fact since most legends have some core of truth- it is very likely. Whether or not that means “Joshua fit the battle of Jericho” is a matter of speculation and controversy. Taking the OT’s tale of “the Conquest” literally is entirely a matter of faith, however. If nothing else, the numbers are WAY off.
As to the word itself- we don’t know where “Hebrew*” (“ibri”)came from. “The Oxford History of the Biblical World” doesn’t have anything on it I can find, and “The Oxford Companion to the Bible” says “the derivation is uncertain” and gives two theories - it comes from the Hebrew verb “abar”- “to cross over or beyond” as in those that came from the other side of the river. OR it comes from Eber, the grandson of Shem. (We do know where the modern word “hebrew” came from, sure- but it’s source is in ancient hebrew and we don’t know where they came up with the word originally). I guess it isn’t impossible that the ancient proto-isrealites knew the word “hapiru” was a term the Egytians often used for them, but … 
Oh, and from what I can see- the term "apiru/Hapiru is Akkadian, not Egyptian per se. The Egyptian client states in the Palestine area corresponded primarily in the Akkadian language.
I am not a strong proponent of the apiru = Hebrew conjecture. I was simply surprised by a claim that it had been disproven (as opposed to “we cannot find evidence to support it”).
Is there a reference to the actual proof that the two are not related?
http://www.biblicaltheology.com/Research/SoggieN01.html
I don’t expect an immediate response to this, since it is a very long article and may take some time to read and digest, but like I said, I think it presents a pretty strong case. Also, it wasn’t written by a bunch of Biblical literalists, so we’re not dealing with guys who are trying to retrofit the archaeological evidence to Joshua. It does address the fact that the word habiru/apiru was used for nomadic/pastoral/outlaw types, but apparently there were the local hapiru, and this bunch of foriegn habiru that came into the country stirring up trouble. It also goes to some trouble of shifting the Biblical record to seperate fact from propagandistic embellishment.
I was tired when I wrote that and I unintentionally phrased it in a misleading way. I meant to say that there has been speculation about a lingusitic connection between habiru/apiru amd Ibri/Hebrew but that it has been disproven that the Habiru should be identified as the cultural group known as Hebrews/Israelites. The way I phrased it made it sound like I was claiming any lingusitic connection had been disproven. That was an accident of phrasing on my part.
I will say that no connection has been proven either and that the cultural group who later called themselves Israelites seem to be indegenous tp Palestine and that there is no evidence that they ever left Canaan. were ever enslaved in Egypt, or ever wandered in the Sinai Peninsula. There is nothing in the archaeology that shows any sort of mass migration into Canaan from the Sinai, and no conquest Canaan.