No. An employee may be entitled to an exemption from a mandatory vaccination requirement based on an ADA disability that prevents him from taking the influenza vaccine. This would be a reasonable accommodation barring undue hardship (significant difficulty or expense). Similarly, under Title VII of the Civil Rights Act of 1964, once an employer receives notice that an employee's sincerely held religious belief, practice, or observance prevents him from taking the influenza vaccine, the employer must provide a reasonable accommodation unless it would pose an undue hardship as defined by Title VII ("more than de minimis cost" to the operation of the employer's business, which is a lower standard than under the ADA).
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You also asked what steps, if any, hospitals may take to scrutinize a requesting employee's beliefs to determine whether the employee's asserted need for accommodation is based on a sincerely held religious belief, and whether hospitals require the requestor to have the support of a religious official. These issues are addressed in great detail, with accompanying examples, in the Compliance Manual, which in pertinent part explains:
Because the definition of religion is broad and protects beliefs and practices with which the employer may be unfamiliar, the employer should ordinarily assume that an employee's request for religious accommodation is based on a sincerely-held religious belief. If, however, an employee requests religious accommodation, and an employer has an objective basis for questioning either the religious nature or the sincerity of a particular belief or practice, the employer would be justified in seeking additional supporting information.
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When an employer requests additional information, employees should provide information that addresses the employer's reasonable doubts. That information need not, however, take any specific form. For example, written materials or the employee's own first-hand explanation may be sufficient to alleviate the employer's doubts about the sincerity or religious nature of the employee's professed belief such that third-party verification is unnecessary. Further, since idiosyncratic beliefs can be sincerely held and religious, even when third-party verification is needed, it does not have to come from a church official or member, but rather could be provided by others who are aware of the employee's religious practice or belief.
An employee who fails to cooperate with an employer's reasonable request for verification of the sincerity or religious nature of a professed belief risks losing any subsequent claim that the employer improperly denied an accommodation. By the same token, employers who unreasonably request unnecessary or excessive corroborating evidence risk being held liable for denying a reasonable accommodation request, and having their actions challenged as retaliatory or as part of a pattern of harassment.