My boss (a gentile) wants me to come visit him and the home office during the middle of the three weeks. He knows nothing about Jews and our Minhagim. Am I permitted to shave during that week.

If not shaving could cause you financial damage you may shave. If on the other hand not shaving for this visit (you can explain yourself) will not affect your relationship with your boss or your work you should not shave. Rabbi Yosef Weitzen