Here's why:
* Proper nouns are specific names of people, places, organizations, or things.
* Common nouns are general names for people, places, things, or ideas.
"Red Cross" refers to a specific organization, the International Committee of the Red Cross and the International Federation of Red Cross and Red Crescent Societies. It's not a general term for any organization that provides aid.
Therefore, "Red Cross" should be capitalized.