Originally Posted by SR116

The carry-on policy is defined solely by the class of travel that you're booked in.

SWISS Economy: 1 bag (max. 8 kg, circumference max. 55 x 40 x 23 cm)
SWISS Business: 2 bags (max. 8 kg, circumference max. 55 x 40 x 23 cm)
SWISS First: 2 bags (max. 8 kg, circumference max. 55 x 40 x 23 cm)

In addition to that, you're also allowed to take a second bag, i.e. a handbag, laptop bag or shoulder bag (max. 40 x 30 x 10 cm).

(Source: http://www.swiss.com/ch/EN/prepare/baggage/hand-baggage )

Since US-sized carry-on bags tend to be larger than the ones found in Europe, there may be chance that they'll gate check your carry-on bag if it surpasses the allowed dimensions.