I hope this message finds you well, present circumstances excluded. I am a licensed employment attorney with over a decade of experience handling matters such as this one. It is a pleasure to assist you today.
Once an employee is on the job, an employer's right to conduct a medical examination is usually limited to so-called "fitness for duty" situations. If an employee exhibits objective indications that he or she is physically or mentally unfit to perform the essential functions of the job (for example, by claiming an injury that makes working impossible), an employer may request that the employee's fitness for the job be evaluated by a medical examiner.
Although the medical examiner can take a full history of the employee and conduct necessary tests to evaluate the employee's fitness, the employer is not generally entitled to all of this information -- only to the examiner's conclusions about whether the employee can work. Many states also impose strict limits on the information a doctor may disclose to an employer or an insurance company without the worker's consent.
Moreover, once the employee is employed, if the employer mandates any medical treatment or physical examination, they are required to pay for the totality of the costs associated with that examination as a condition of employment.
So, no, what they are doing is not legal and you (or someone else) needs to let them know that in writing. The best way to tell them this without drawing negative attention is to request they provide legal support for their decision to require this yet not pay for it. When they research the matter, they will find out that they cannot do such a thing.
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Best wishes going forward!