I don't think you can get in trouble for using such a symbol in a game, due to the fact that copyrights and trademarks only apply in the setting they reference. You cannot create a health NGO with the red cross symbol, but you sure can create a clothing store.
Besides, the red cross is much too simple, and as such any claims they might have are worthless. Beyond that, games are used to certain tropes, and such tropes are protected as public domain. Also, since they are an international humanitarian organization, it's extremely unlikely they'll take action against game developers.
In addition, they also apparently want to patent the "red crystal" and "red crescent", and since green/blue/orange crosses are being used worldwide as alternatives, it's not hard to imagine a similar misguided movement attempting to protect them. As much as I hate bringing up the slippery slope argument, I can see this becoming a "no colored iconography of any kind on white backgrounds" law.
The red cross society may plead and beg, but their request is quite frankly absurd, and game designers should purposefully ignore them.
Disclaimer: I'm not a lawyer, and as such my "legal" advice is worthless.