It was founded earlier than that. I believe it was witnessing the horrors of Solferino that put the idea in his head (I remember pictures of lots of soldiers in blue uniforms and others in white ones from an old Blue Peter Annual article on the subject).
Yep, that was it, Solferino. Same battle that Franz Josef II witnessed in person. He was so horrified by the battle that he spent the rest of his days wearing a Corporal's uniform, and sleeping on an army cot.
If I recall, there is a reference to "red cross" wagons being used during the Austro-Prussian War of 1866. Not one hundred percent positive, but that sticks in my mind as the earliest reference I'm aware of.
-Doc