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As you may already know, if you are in a state that follows the doctrine of "at will" employment (as all states do but Montana) then you can terminate an employee for any reason not amounting to discrimination on the basis of a legally protected trait (race, religion, gender, etc.) or retaliation for engaging in certain forms of legally protected conduct (filing a wage claim, taking FMLA leave, etc.). It doesn't matter whether the basis for termination is fair, reasonable or even true.
Since you can terminate with such impunity, it really doesn't matter whether the employee claims they were fired even if they resign. Being fired on its own won't mean they have any sort of legal claim. They would need to prove that your MOTIVATION for firing them was a legally protected trait or activity. As you indicate that you had to discuss serious matters with the employee, I am guessing you would be able to point to those reasons as the basis for this employee's "termination" even if they could somehow establish that they were fired. That right there is enough to defeat any lawsuit.
The distinction between resignation and termination most likely only matters with respect to the employee's right to collect unemployment benefits. UI benefit eligibility is judged by a different standard--whether the claimant is unemployed through no fault of their own. A claimant who quits their employment will usually be denied benefits since they made the voluntary decision to become unemployed and are thus unemployed "through fault" of their own. However, a claimant who is fired will be eligible for benefits unless the employer can establish that they terminated the employee for "misconduct," defined as conduct evidencing an intentional or reckless disregard for the interests of the employer.
I hope that you find this information helpful. Please do not hesitate to let me know if you have any questions or concerns regarding the above and I will be more than happy to assist you further.
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