Your goal is to get Canadian permanent residency? If you are claiming in Canada to be a resident of Canada while claiming in the U.S. to be domiciled in the U.S., I think you should consult an immigration firm that handles both U.S. and Canadian immigration issues (or two lawyers, one for each country).
Given that you can live, work and attend school in the U.S. on a H-1B visa, it would make sense to maintain H-1B status.

