tulamide
Sorry, I don't know the laws in Germany. I'll take your word for it. In the U.S., it has to be proven that the red cross symbol on white background was used as intent to misrepresent the ARC or harm it in some way. That would be very difficult to do unless it's obvious and deliberate, like using the words "American National Red Cross" along side the actual symbol or painting the symbol on your car to fool other's into believing you represent the ARC or maybe selling a product misleading people into thinking it was endorsed by the ARC. Those are proof of deliberate acts that can harm the ARC.
The ARC charter of 1910 clearly states that using the red cross is against the law if it's used on vehicles, vessels, aircraft, buildings or other structures, or upon the ground. No mention of use outside those conditions, as far as I've read. That leads me to believe that they are not so much concerned about protecting a red cross from being used as they are trying to protect the symbol that represents the ARC from being misused. What I mean by that is the red cross on a Knights Templar garment does not mean the same thing as the red cross for ARC. Red crosses other than those meant to represent the ARC and Band-Aids are fair use in America. The example I linked to was meant to prove that no one is being prevented from using red crosses and not as an indication of law.
If you are still worried about using it in a game, don't. It's not worth the headache. Use a heart symbol or something else to represent health.
--Edit--
I forgot to attach my source. Please refer to the final page of the charter. It's illegal to use the red cross to misrepresent the ARC, otherwise it's ok to use it in the U.S. without penalty as stated above.
Congressional Charter of the American National Red Cross