In this case, the employer is a medium-sized ENT office. Some employees are young single mothers who don't have the resources to comply with this. The question also applies to non-medical employers, of course.
Depends on your definition of "require".
In case of at-will employment — where the employer can fire an employee for any or no reason (other than that being discrimination of a member of the protected groups) — it would be perfectly legal for the employer to fire an employee who does not comply with that request.
But that aside, no (unless such testing was a term of the employment agreement, be it written or verbal). No party to a contract can require the other party to perform what the contract terms did not include.