Surely, "Chinese man" and not "China man". The latter sounds close to chinaman which I think these days would be considered derogatory.
Florida man does not carry this cultural baggage.
Thank you for bringing it up. I also don't understand why some people get so offended when they are told that certain words can cause harm to others.
I always try to be respectful and if someone tells me that a word or a name hurts them, then I just won't use it around them.