The rule can be found in the ASA directive No. 1
https://asa.ch/it/biblioteca-online/direttive/
According 243 and the example given you must get Swiss plates 12 months after taking up the Swiss residency. (Or store it on private property and not use it on public roads which is also o.k.)
Note 1: To legally use an US plated car on European roads you will need a valid third party liability insurance in form of a frontier insurance/Green Card insurance. (Available at a Swiss customs offices
https://www.ezv.admin.ch/dam/ezv/de/...herung_dfi.pdf or
http://www.tourinsure.de/en/internat...nce/car/europe ). This is quite costly and getting Swiss plates an proper insurance is usually cheaper. Your US liability insurance is not good enough and is not accepted!
Note 2: Your driving license is independent from the car. You need a valid driving license to drive the car. Your US license will be void to drive on Swiss road 12 months after taking up residency. It is best to exchange it before the 12 months are up. And yes, you can drive a US plated car with a Swiss driving license.