Could you automatically become a BSA lifeguard if you are American Red Cross Lifeguard certified?

No, one doesn't get two certifications for just one class. I am a BSA aquatics instructor and am also Red Cross certified, and while you have to be seperately certified, by taking the Red Cross lifeguard course, you will have no problem going through the BSA class also. They are very similar, but still have their own tests and requirements.