Changing the color of the cross doesn't make it any less (or any more) Christian, so it isn't a religious thing.
As I mentioned, it isn't just in the U.S. that the Red Cross has come down on others it perceives as "infringing" on the symbol, so it's not just the
American Red Cross that gets involved.
Actually, TW, I think the (International) Red Cross contends that it is
because of the organization's efforts and impact that the symbol has become inextricably connected with health and hospitals, so it feels entitled.
In a sense, the organization is trying to keep its trademark from becoming universal -- generic -- and thus up for grabs by anyone. It's a big issue in the business world (and don't let anyone try to sell you on the notion that the Red Cross isn't a business). Decades ago, the Bayer company owned the word 'Aspirin' as a trademark -- the pills were even marked by that word instead of Bayer in a
cross (there it is again)--
Code:
A
S B
P A
A S P I R I N vs. B A Y E R
R E
I R
N
-- but (at least in the United States, I'm not sure about elsewhere) the word became a generic term and Bayer lost that exclusivity. You've probably seen ads from Xerox/Kleenex/Band-Aids and other brands differentiating between photocopying/facial tissue/bandages (plasters to you Brits) or whatever, and what belongs to them. They don't want to lose the brand-recognition that they've spent so much money and time to build up -- though it's also a result of the efforts to
create that brand-recognition that so many of us, for example (again, at least in the U.S.), think/say 'Kleenex' regardless of who made the tissue.