Your first sentence was broader than that. Far as EU, they're called registered designs. I don't know much about them past that they exist. They're also in Hague Agreement of WIPO which Switzerland is a party to. Apple's attack on Samsung in Germany was about them copying the interface and style. Situation isn't as clear cut to me given the above.

Even if nothing applied in Switzerland, the countries supporting such things might block their local presence there under grounds of violations. This risk actually is why some companies, esp Chinese, won't do business in the U.S.. If this played out, they'd be blocked from serving customers in U.S. or any nation that upholded the claim against them.