There are quite a few jobs/professions that circumscribe one's personal choices more than requiring flu shots ever could. In many localities, if you're a police officer, fire fighter, etc., you have to live within the municipal boundaries. As a lawyer, you're required to take on court appointed cases, even if it's in an area of law you've chosen not to practice because you felt ethical qualms about representing certain types of people. Should a teacher who is a firm creationist be allowed to refuse to teach the theory of evolution? A pharmacist refuse to distribute the morning after pill? A police officer refuse to carry a gun? The wide array of workers who are required to take TB tests refuse to take them? (That also breaches the skin.)
I think that, with certain jobs/professions, you're aware (or should be) that you may be required to do things that you would prefer not to do. It's up to you to decide whether it's a dealbreaker.