Why is the red cross called the red cross?

The founder of the International Red Cross and Red Crescent Movement is Mr. Jean Henri (or Henry in the English spelling) Dunant who is a Swiss. After he decided to found this Movement, the Swiss government helped Mr. Dunant a lot. While they were thinking about the protective symbol for hurt soldiers and medical assistants, they want to show their thank to the Swiss government. Therefore, they reversed the colour of the flag of Switzerland and named it the "Red Cross".