What is the meaning of Switzerland's national flag?
The Legend of the Flag with a Cross
But what are the origins of the white cross on the coat of arms of Schwyz? During the middle ages, Switzerland was first part of Charlemagne's great continental European empire (around A.D. 800). His sons split the empire in three parts, the eastern part (including Italy, the Alps and Germany) was called Holy Roman Empire of German Nation.
When the German emperor once went to Italy around 1230, soldiers from Schwyz accompanied him. It is reported that he was very pleased about the support they gave him and granted them the privilege to add a white cross to their red banner.
An old legend says that one of the last Roman emperors, Constantin, used a banner with a cross symbol when he defeated his rival Maxentius in 312. It is very likely that the cross symbol was taken over by Charlemagne and his successors as a sign of their leadership over the "Christian continent".
The privilege to use the Emperor's cross on their banner was not exclusively granted to the people of Schwyz, however, and so you may find many other political entities using a similar coat of arms, for example the Dukes of Savoy or the city of Vienna (capital of Austria) and Danmark - all showing not only the same symbol  (white cross on red background) but even in the same colors, while Finnland and Sweden use different colors and the United Kingdom's Union Jack and the Dominican Republic have different colours and some additional features in their flags.
Who "Founded" Switzerland's National Flag?
The answer depends on the focus:
If you go right back to the very origins of Swiss troops using the white cross on red banners, this was a privilege granted by German emperor Friedrich II., so you would suggest that he "founded" the Swiss flag.
But if you are asking for the person who convinced a majority of Swiss politicians to use a common national flag instead of a variety of cantonal banners in 1840, when the white cross on a red square emblem was officially introduced as the Swiss army flag, this was Guillaume Henri Dufour, who later became general of the federal troops in the short Sonderbundskrieg (civil war) in 1847 and co-founder of the International Committee of the Red Cross in 1864. Of course, it was also Guillaume Henri Dufour who proposed the red cross on white ground as the emblem and name for the Red Cross.
Facts about Switzerland's Flag