Early modern Switzerland, particularly in the west around lake Geneva and Bern, was a centre of reformed Protestantism, and therefore was seen by English puritans as a place that would welcome and protect them. Many also headed for Amsterdam, which was also a centre for reformed Protestantism, but the king's agent Sir George Downing succeeded in arresting several regicides there in 1662 and deporting them to England for trial.

Switzerland was seen as safer, therefore, and many headed for Geneva, which was viewed, in the words of the regicide Edmund Ludlow, as a city 'renowned for liberty and religion'. It was in the canton of Bern, however, that the regicides received official recognition as being persecuted for their religion, and many received letters of protection from the canton's senate and settled in places like Lausanne and Pays de Vaud which were under the jurisdiction of Bern at the time. Some were also advised to move to Vevey, which was seen as quieter and safer.

Switzerland was therefore somewhere that many regicides saw as a place which would sympathise with them politically and religiously, and where they could be protected by local authorities. Even so, Switzerland still wasn't entirely safe, as the exiled regicide John Lisle was assassinated in 1664 in Lausanne by royalist agents.

Source: The English Republican Exiles in Europe During the Restoration by Gaby Mahlberg.