Could phentermine give you a false positive on a urine drug screening?

Maybe. It is possible on some drug screens for Phentermine to show up as amphetamine. I usually warn my weight loss patients of this and write them a letter for their employer. Your doctor should be able to write you a letter in case this happens.
Yes. It can cause a false positive amphetamine, but confirmatory tests, which should always be done when testing is for forensic purposes (probation/parole, job-related, social services). However, many people doing drug testing are not trained in it, and often think that results of a screening test are definitive. You have the right to demand confirmatory testing with gc/ms or other means.