Everything I've heard up to this point suggests no, you can't - my guess is because one of the conditions of the BOWP is that you have to "currently be in Canada", and applying at the border involves leaving Canada.
I certainly asked CIC about this when I first called about the BOWP, and they assured me it wasn't possible. Others may have had different advice or experiences, however I haven't heard of any cases where this plan has worked (otherwise I'd be doing it myself right now!)
