H1B in the US- Legally, you need to have your valid Canadian Passport going along with your H1B all the time. The border officer would say ," Your H1B is fine but your passport is expiring. If you don't get your new passport in Canada, you would have a period in the US without a valid passport. So you need to get your new passport." If you can stay in Canada and wait for your new passport to be issued, then everything will be fine. If you can't, then you'd better not take the risk.