Reddit reviews A Very Civil War: The Swiss Sonderbund War Of 1847

The connection is even closer. The Swiss flag design was adopted in 1841 upon the recommendation of Henri Dufour of Geneva, who was going to win the Swiss Civil War for the Federal/Northerners in 1847. He is Switzerland's US Grant with the little difference that he managed to knock out the Swiss South (agriculture and conservative Catholics, no slaves in Switzerland) in two lightning campaigns that blasted away the rebel army with artillery but spared civilians ("with malice toward none") in one year. After the war, Switzerland changed from a confederacy of states into a federal state with its two chamber parliament based on the US constitution. Fearing a powerful executive, a seven-person executive committee structure was chosen. To placate the Southerners, the name "Swiss Confederation" was retained despite being clearly a federal republic, which causes almost as much confusion as the US slogan "a republic not a democracy". An excellent book about the Swiss Civil War, sadly out of print, is Joachim Remak's A Very Civil War.

A few years later, in 1863, Dufour became one of the five founding members of the Red Cross and its first president. Four of five were Swiss citizens. So when the decision to chose a flag, retaining imagery from the Swiss flag was not a strange idea. It was also fitting as Swiss neutrality which had been officially in place since 1815 meant that the Swiss flag would never be used by belligerents, thus an easily recognizable sign of peace and succor.

Incidentally, especially in the US, many first aid kits and emergency services reverse the red cross, displaying a Swiss flag.