Amy, yes it will count towards clinical experience.
I've volunteered with the Red Cross for about two years now and it's been great. I highly recommend it. I also volunteered in a large hospital's ER and the Red Cross volunteering was much, much better. I got to see how an ER operates at the hospital, but I wasn't allowed to do anything. With the Red Cross there was tons of hands-on experience (and free CPR, AED and first aid training).