Dual US-Swiss citizen with family emergency and expired US passport

Well you cannot be refused entry to the US, but on the other hand, with the Swiss passport you need an ESTA to board the flight, which will likely be denied as you're a US citizen (but if not, you're good to go - in this case only present the Swiss passport to check-in staff, and only the expired US passport to US border control).

So my advice: apply for an ESTA first thing you do and see if it gets approved.

Another option is flying to Canada or Mexico and continuing to the US by land. For Canada, besides your Swiss passport (which requires an eTA), you can also use a US birth/citizenship/naturalization certificate (which doesn't require an eTA).

Yet another option is to get an emergency passport from the US embassy in Bern.

I see only two realistic options to quickly get to the US without a valid US passport:

1. Get an emergency passport in Bern for $110. This is the logical option.
2. Fly to Canada or Mexico using your Swiss passport and then cross the US border by land. Bring all the documentation you can -- expired US passport, birth certificate, other proof of citizenship, Swiss passport, driver licence, other proof of identity. If you can prove that you are a US citizen, US authorities are legally required to let you in. But be warned that they are not required to be nice to you. It is technically illegal, but there is no penalty. So expect a lot of unfriendly questioning if you try this option.

Flying to the US without a valid US passport is generally not an option. You are very unlikely to be given a boarding pass if you don't satisfy the US entry requirements. The airline employees can be stricter than the US authorities, because the airline can be fined and is liable for the costs of returning anyone who is not admitted on arrival in the US (and you have no legal "right" to get on their plane).

To use your Swiss passport you need an ESTA. I'm not sure if you are eligible for an ESTA as a US citizen; some say it can be done. It's a risky option, but I guess you could try it. Just be absolutely sure that you don't make any false statements on the application.

Always keep your passports up-to-date for such emergencies. You never know when you are going to need to travel.

You can likely get a rushed US passport renewal, but flying on your Swiss passport and presenting your expired US passport as evidence of your US citizenship is probably the best course of action. Your airline is going to require you to have an ESTA to fly to the US, but as a US citizen, you're not entitled to get one, so be sure to show them your US passport if asked, too. The more recently the US passport has expired, the less trouble you'll have, but you are entitled to US entry - it's just going to be an issue of whether the airline believes you'll be admitted to the US or not. (They have to fly you home for free if you won't.)