If you have a valid green card and a valid national passport, you do not need to apply for a visa to Switzerland and you can stay for 90 days. However, take care that both green card and passport must be valid atleast 3 months after leaving Switzerland.
No, they don't need a visa for Switzerland. This is spelled out in the Swiss Consulate website and make a copy of that for your convenience.
I think US citizens have advantages when traveling to different countries since they are allowed legal entry even with the use of a US passport.