Do I have to pay back my employer if they've paid me too much?

Your question presents a number of issues.  First,you are correct - if you are salaried, in general your pay cannot depend on the number of hours worked.  It is unusual, but not impermissible, for a salaried employee to receive overtime pay.  The better question is are they really treating you as an hourly employee (shorting your pay for hours not worked) but claiming that you are exempt from having to be paid overtime when you work more than 40 hours in a week.  If this is the case, find a qualified labor attorney right now. Second, an employer may request an employee repay amounts paid in error.  If the employee says "no" he may or may not have to pay it back (see below) but most likely he will be fired. Third, an employer may not deduct any amounts from an employee's paycheck (including past overpayments, but not including taxes and child support) unless the employee has given written authorization for the deduction.  Even then, the deduction may not reduce the employee's wages to an amount below minimum wage.  Many employers require employees to sign such authorizations at the beginning of their employment, particularly for things such as lost tools and uniforms.  However the authorization must be specific enough to cover the claimed deduction.