My advice is to have the medical. If it is a requirement of the job (just like flying,) then put up with it. If an outside contractor is employed by your company to undertake these medicals then any concerns about your personal data being on company computers seems fairly unlikely. If you don't like being watched "peeing into a cup" then ask to do it in private or turn away from the doctor. I can't help feeling you are worrying a lot about very little.
Of course you can refuse to undertake the medical, although you seem to indicate that it is a contractual requirement? Ultimately the choice is yours, but it still seems the easiest route is simply get it over with.