Jenyar,
From what I can determine the red cross was never intended as a Christian symbol. It is just the inverse color of the Swiss flag, although that had Christian roots. But the modern movement has maintained a strict policy of religious independence and clearly state that the red cross is not an indication of religion. They are currently in negotiations to instigate an entirely new neutral emblem that will not be confused with potential religious connotations.In that light, maybe it was a mistake to opt for the "red crescent" to make the movement more acceptible outside christianity - it could have been a more neutral symbol, and it reinforced the christian connotation with the simple red cross - but unfortunately symbols are rarely without emotional meaning.
Well that is not clear. According to his biography he was certainly born into a Christian family, and it would appear somewhat like me, that he actively pursued religious interests when he was young. But the founding of the Red Cross seemed to have had nothing to do with Christianity or religion. This began entirely for humanitarian reasons. That he was probably a Christian may or may not have had anything to do with these actions. I could not find any reference to his possible religious beliefs playing a part in the founding of the Red Cross.The Red Cross was established by a devout Christian, Henri Dunant.