Police confirmed 17 passengers and three crew members on the JU-Air flight were killed, with the dead aged between 42 and 84.
The victims included 17 Swiss nationals and a family of three from Austria.
Ju-52s were used as a Luftwaffe personnel carrier in the Second World War.
But the aircraft was most notorious for serving as Adolf Hitler’s personal plane between 1932 and 1939.
Heavy winds and the hot weather may have played a role in the crash, which happened on the western side of Piz Segnas mountain, 8,000ft above sea level.
JU-Air was founded in 1982 and specialises in sightseeing and charter flights in three former Swiss Air Force Ju-52 aircraft.
JU-Air chief executive Kurt Waldmeier said: “Yesterday was the worst day in the 36-year history of JU-Air. We have all suffered a very great loss.”
He added that the aircraft had been serviced at the end of last month and no defects had been found.
The tragedy was the first fatal crash the company has been involved in.
It has suspended all flights until further notice.
Daniel Knecht, head of the Swiss Transportation Safety Investigation Board’s aviation division, said: “One can ascertain that the aircraft hit the ground nearly vertically at high speed.”
An investigation is expected to be complicated by the lack of a black box flight recorder.
Police said that they were not aware of any distress call and had not yet determined the cause of the crash.
Saturday’s tragedy happened just hours after a family of four, including two children, died when their small aircraft crashed into a forest near the town of Hergiswil in central Switzerland.
It is understood the family were flying to France.