There is not statute providing employers with this right generally. However, under the employment at will doctrine, employees are generally free to set the terms of employment, including reasonable policies documenting an employee's absence or release to work. The request should normally be made pursuant to a written policy which has been previously provided to all affected employees, so it is clear that the employer is not singling out an employee on a discriminatory basis for less favorable treatment.
For purposes of unemployment benefits, the burden is on the employee to prove they had good cause to quit where they voluntarily quit their job normally, so I would expect the decision to be reversed on your appeal, but it would be best to retain an attorney to represent you.
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