Hospital volunteering is just one of several hoops that you have to jump through, regardless of its relevance with respect to a medical career.
I would say that clinical experience is required, but not hospital volunteering. I've never volunteered at a hospital in my life, and it didn't hurt me.
By clinical experience, I mean enough experience to have some idea of what you're getting yourself into. It's up to you how much is enough, I think the amount that people suggest is necessary on SDN (hundreds of hours or w/e) is total BS, but to each his own. It's the value of an experience, not the amount of time.
OP, if you are truly interested in volunteering in a hospital (not because you think it'll help you get into med school, but because you have some free time and you really want to volunteer simply to volunteer), by all means go and sign up and do whatever stuff they ask you to do. If you're doing it just because you think it'll help you for med school, then I wouldn't do it.
If you can find a physician to shadow (which you can do in high school), that would be much more worthwhile. You learn a lot more, and it's more fun.
My point is, do stuff that'll make you happy and that'll help you make an informed career choice (notice that I didn't say the "right" career choice, because there's no such thing). Don't do things because it'll help you get into medical school, do them because they are enriching, or because you will enjoy them.