H1B and travelling while it is still in the approval process

Hi. I have tried google and I can't seem to get an answer. Husband has been offered a job with a US company. It is our understanding that they intend to peruse an H1B visa for him. We are unsure if he should give notice to his current Canadian employer until he actually has his visa approved and in his hands. However, he is suppose to be travelling to the USA at least two times on business in the next few weeks for his current employer. What should he do? Can he even travel to the states legally while an H1B is in process? What does he say to border patrol? I'm suppose to go with him at the end of next week to look for housing... what should I say to border patrol? HELP!!

Until the H1 is approved AND he formally changes his status at the border, he enters US as a business visiter, so that is not a problem.
You are simply accompanying him on his business.

He should NOT give up his job until his H1 is approved.
Any reason they aren't simply getting TN instead?

Thanks for the quick reply. I don't know why they are doing an H1B over the TN... I assume because it's easier to adjust status? I really have no clue why as he qualifies for a TN under a very clear cut category. He will be entering before me and meeting me.... When I go to the POE I guess I will just say he has been offered a job and I'm going down there to meet him and check out the area. Thanks for your help.