?
If so it would be easier.
We (both non-EU) did this for a couple of stints - my husband worked for a couple of years in the US and in London, I (an unemployed frivolous dilettante housewife ) stayed here. The HR folks at his company worked out the permit details.
A key issue was that he was paid through the Swiss office, he traveled back to CH for work with some regularity, and that he kept the 'center of his life' (me and the dogs, I suppose - and our main residence ) in Switzerland.
And most importantly, we continued to pay taxes in Switzerland.
As long as we kept paying Swiss taxes the good folks at the Gemeinde and cantonal Migrationsamt were happy to let us keep our residence status.
(Of course he had to pay US taxes, too. That was fun... But it was the price of keeping our residency in Switzerland. )
If your husband is working in the US for a different company than his current one, or if he will be wholly a US employee then the situation is different - and you'll need to consult a lawyer, as others have suggested.
Hope it all works out for you.
Just to be clear, I am interested in paying taxes in Switzerland and to contribute to the community in every way that I can while I reside here with my family!
As for my husband's work, he is an academic researcher who has been collaborating with his European, Asian and American colleagues for 25 years. The idea isn't about ownership or profit. It's to join heads and hearts to come up with ideas that contribute, somewhat irregardless of nationality or money.
He has been offered a position in Switzerland that honors the work he does here and would provide us the opportunity to extend our residence permit here. However, in accepting this temporary situation, he would potentially reduce his chances of securing permanent employment at the level he is interested in achieving. We could resort to accepting the offer, but we are hoping to maintain our residency here through my European citizenship until he secures the kind of position he seeks in CH. He will continue with his U.S. based research and work here as an independent collaborator/researcher 12 weeks interspersed throughout the year.
I am sorry if I seemed to paint a picture where we seem to want to slip into Switzerland unfairly and avoid taxes or social responsibility. Thankfully, my experience here has been that we are welcomed by our Swiss and international neighbors and colleagues.