Expired destination (U.S.) passport; unexpired dwelling (Israel) passport

Unless you can get a same-day passport or transportation letter from the US embassy tomorrow, the simple fact is you're not flying tomorrow.

Timatic, the database used by Airlines, states:

Passports and other documents accepted for entry issued to nationals of USA must be valid on arrival.

And of course the Israeli passports are not an option as your kids lack visas.

While US citizens cannot be refused to entry to the US, airlines still follow Timatic, so again, you'll have to reschedule your trip (do note that you can get the airport taxes for the unused flight back).

Note: To save the costs of a same-day passport/transportation letter, you can fly to Canada and cross by land. For Canada, Timatic states:

Passport Exemptions: Nationals of the USA with a recommended proof of citizenship such as a:

• US birth certificate; or
• US certificate of citizenship; or
• US certificate of naturalization.

So that is an option too, and possibly cheaper

You are required to leave Israel on your Israeli passports (as I am sure you know). And if you can get to the US border, as citizens your children must be allowed in. However, there may be an issue with boarding the flight without a visa. Given that the embassy is unlikely to be able to help you on such short notice, I think your best chance is if the children were born in the USA, which would be in the passport. That might be enough to persuade an airline clerk that they are citizens. Similarly, if the children were born in Israel, you could try bringing the Consular Report of Birth Abroad, and seeing if that will persuade the airlines.

Good luck.