We can't install customer supplied parts. As the professional, we become liable. There are tons of articles about this subject. Shops have gotten sued after installing customer parts that fail and they lose. We as the professionals are liable for the work we do, parts do not matter. I do not allow in our shop not only for this reason, but because parts & labor are how we profit in order to stay in business. We only use certain brands for certain jobs. We over time have come to find what good brands are and what bad brands to stay away from. Why? Because we are the professional! How many brakes, ball joints, calipers, Mass Air Flow Sensors, etc have we installed in our careers? We know what fails and what doesn't. Stand up, be proud of your profession and let them know exactly why we can't adhere to this practice. We deserve to make a profit and make a living for the work we do.
Still haven't even thought about taking my own steak to a restaurant. I feel great to pay others for the services they provide for me that I know nothing about. Home HVAC, not my thing, but I know who to call. Plumber!!! Got a company for that too.