Your lawyer is totally wrong

As a Canadian citizeh, nomatter where you were born, you do not need any visa to visit USA, to work in USA. What you have with H1b is the H1b status not visa (similarly TN status not TN visa). You only need to show your H1b approval document (797) first time when you go through the border by either land or air, or sea. You then will get the I-994 with H1b sign on it. After that, your passport with that I-94 is more than enough for you to enter USA. Hope this helps.