There was a book published in 1969 "If Israel Lost the War" which had Israel being defeated in the 1967 war, which begins with a surprise attack by the Arab air forces, reversing the opening events of the OTL war. I agree that Israel losing massively is highly improbable given the relative strengths of the various militaries, the level of competence of the Arab militaries, and the rather limited coordination. Having said that the POD is that the Arabs do win, and the original poster is not asking how (handwavium applied) but rather "now what?"
American intervention before all of Israel is occupied is highly questionable. The USA in 1967 has no ground forces available for rapid deployment to Israel, and is neck deep in Vietnam. The only US forces potentially available are naval forces such as a carrier battle group and possibly a Marine amphibious ready group with the 6th Fleet. Air power from Europe is potentially available, however this requires the countries where such forces are based, as well as any NATO countries they would overfly to agree to land based air forces being so used, which is not a given. Once the decision to intervene in some way is taken, and this may not be so rapid, the naval forces have to motor to the Eastern Med. The problem with US airpower being used in such a fashion is how to direct such strikes. The USA and Israel have not practiced such coordination, communicating between Israeli FACs and USN/USMC (and potentially USAF) aircraft is going to be very ad hoc and problematic. There WILL be tons of refugees fleeing (think France 1940) and all of this makes blue on blue or significant civilian casualties due to poor communications a severe problem. Deeper strikes against Arab rear areas/logistics are less difficult in this respect but would the USA be willing to strike across borders into the pre-war Arab countries proper. Hitting targets in Israel might be something they could get away with, but hitting targets across prewar borders risks the Russians getting involved to "protect" their proteges in Syria and Egypt.
Once the Arab armies and irregulars/militias are well in to Israel, and intermixed (deliberately or not) with the civilian (Jewish) population there is only so much US air power can do. If there is an ARG that can be deployed, US Marines with remaining Israeli forces and the USN floating offshore could probably make a perimeter around a Tel Aviv - Haifa coastal strip. While the "liberation" elements would not care about US forces, I doubt the Arab countries would attempt to overrun US manned positions - killing Marines might just piss off the USA to cause the loss of their gains.
In the "best case" scenario of such an enclave being protected (at least temporarily) by the USA, you will end up with all of the surviving Jewish population of Israel (roughly 2.5 million) and potentially some Druze and Christians crammed in to this coastal strip which would have had a population of 500,000-750,000 as a rough estimate. Some of that population would be Arab/Muslim which would most likely leave as soon as possible (voluntarily or otherwise). This area has limited industry, almost no agriculture, and even once you house all of these people permanently (and they will be in tents for years), no room for any significant industrial or agricultural growth. It is surrounded by an implacably hostile coalition, and subject to constant terror raids - and has no ability to retaliate meaningfully. Long term finis Israel.
In the wake of this, and as the US turns away from Vietnam after 1967, does anyone think the USA will make a real commitment to protecting this unviable ministate with US forces stationed there absorbing casualties from terror raids? Would the UN blue helmets be of any use (like they were in the former Yugoslavia or Rwanda)? IMHO this rump Israel becomes a temporary holding camp as the Jews, and others who don't care to live in "liberated" areas scramble to find some place to go to. At some point I expect the Arab countries will either simply walk in, with or without an ultimatum giving the Jews (and others) a certain time to leave "or else".
If the "international community" tells the Arab countries involved that they need to negotiate a peace settlement and withdraw to (new) borders the response will be "this is liberated Arab territory, we are not giving back an inch - go ahead make me." There the discussion will end. The Saudis and other Gulf oil states could care less about the Palestinians (they don't) but they will use the oil embargo threat and for sure the Europeans won't back any serious measures against the Arab conquerors. Again, with the USA neck deep in Vietnam, unrest over this beginning to rise in 1967, and the Russians growling in support of their proteges, there will perhaps be lots of hot air, serious tut-tutting and that's it.