the flag of Fribourg, a large canton in the West of Switzerland, is the same as the Templars flag - white below and black above
The flag of Fribourg started wearing the black and white coat of arms only in 1714. So its origin was not due to the Templars. Before 1714, the coat of arms of Fribourg was three white towers on a blue field. The latter had been used since the 13th century.
The national flag, red with a white cross, bears a greater resemblance to that of the Hospitallers (Is there any connection between the Order of St John and the House of Savoy?)
The Swiss national flag, as you said, looks like an inverted version of the Templar cross and colors, somehow different from the coat of arms of the Hospitallers who wore a white cross on a black field. Both the Templars and the Hospitallers had possessions in Chambery, but it was after the town became the seat of the Counts of Savoie in 1232, only one year before Thomas I de Savoie died.
The Count did not seem to have been a direct benefactor to the Templars. Perhaps Thomas I de Savoie decided to use a crusader-looking logo because he came from a family of crusaders. His grandfather, Amedeus III, took the cross in 1147.
So if the Swiss flag could be the result of Thomas de Savoie's link with the crusades, it cannot be interpreted as a result of the Templars fleeing to Switzerland in 1307.