Is it permissible, Islamically, to require an employee to urinate in front of another person to provide a urine sample, under the pretext of ensuring the urine exits from his penis?

It is not permissible to expose one's awrah (private parts) except to one's wife or female slave. Necessity is an exception to this rule, but what the questioner mentioned is not a necessity, as it is possible to ensure there is no cheating through other methods, such as searching, allocating a special place, or monitoring without seeing the awrah. If the doctor insists, he should be advised, then the matter should be escalated to the authorities. If they do not respond, it should be escalated to the Shaykhs and scholars.