Hi perplexed,
I believe that what you've been told is incorrect. As far as I know, Canadians need an F-1 visa, and in order to get one, you need an I-20 form from your school. If you don't have a visa, you'll be living in the states illegally.
Now that I've heard your story, though, I'm worried about getting the I-20 as well--I'm using TERI (CanHELP) too, so hopefully my school will allow that as proof of funding. They didn't seem to have a problem with it when I mentioned it during my interview. I don't know why any school wouldn't take it--it doesn't make sense to me. After you've been accepted and processed by TERI, your tuition and living expenses are guaranteed to be covered (according to your school's budget). That's terrible that your school won't recognize this. Is there any way, like musikus suggested, that you can talk to someone "higher up" and maybe get their policy changed?
Good luck to you.