You jest, but the answer is probably “yes.” Influenced by Free Masons, following the Calvinist-Catholic wars, the Swiss emerged as formally neutral in European conflicts (including to their eternal dishonor, WWII). Hence, the Swiss flag became a symbol of neutrality, including medical workers in battlefields. Thereby, it became a symbol of also of medicine in general.
You're joking, right? We're talking about the freaking CROSS here. It didn't originate with the Templars or the Swiss. It began at Calvary. And it became an emblem for medicine in an era when most nurses were nuns.