The Swiss newspaper 20 Minuten reported Miller, 10 band members and a backup driver were not seriously injured.
The newspaper said the bus overturned about 11 a.m. on the A2 highway between Erstfeld and Fluelen. The cause of the crash had not been determined.
The 53-year-old, Grammy-winning musician provided no details on his website, but his homepage stated: "Thanks to everyone for your thoughts and prayers."
The band had played in Monaco Saturday and was on the way to Hengelo, Holland, for another performance Monday.