Quick summary:

 1. The CBP (Customs and Border Protection) and DHS (Deparment of Homeland Security) are quite clear about the entry requirements: US citizens need a valid US passport to enter the US when arriving by air. However, they are also very vague about what happens if you don't have one. 
 2. As a  US citizen, you can't be denied entry, which is in direct conflict with the requirements above (See for example https://law.stackexchange.com/questions/16654/can-us-citizens-be-denied-entry-to-the-us). In practice, most likely you will get in, however the CBP can make the process miserable for you. 
 3. No airline will let you board a flight to the US with an expired US passport and they have every right to deny you boarding if you don't have proper paperwork. 
 4. You can try to apply for an ESTA using your Swiss passport, however during the ESTA application you have to list all citizenships that you have. As a US citizen your application will be denied, so you would have to lie on the application. See https://esta.cbp.dhs.gov/esta/application.html?execution=e2s1