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Your employer is correct that Virginia is an "at will" employment statement. This means that an employee can be terminated 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. It doesn't matter whether the basis for termination is fair, reasonable or even true.
This doesn't mean you have no claim, though. It is illegal to terminate an employee in retaliation for their use of FMLA. Given that other employees were not disciplined for making similar mistakes and that you were removed from phone duties for no good reason shortly after renewing your FMLA, you can certainly try to make the argument that your employer was retaliating against you for taking FMLA leave.
Therefore, it would be a very bad idea to sign any documents stating that you resigned. First, you didn't resign and it is never wise to sign something that isn't true. Second, sighing such a statement will essentially preclude you from claiming wrongful termination, since you will be admitting that you weren't terminated. Finally, signing such a statement may preclude you from collecting unemployment benefits.
You can pursue an FMLA retaliation claim either by filing a lawsuit in civil court with the assistance of an attorney or filing a complaint with the Department of Labor. Hiring an attorney is preferable because the DOL is understaffed and does not "represent you." They are a neutral agency that will investigate, but they have very limited power to actually resolve disputes.
Lawyers take viable FMLA retaliation cases on a contingency fee basis. If you don't know, a contingency fee arrangement is one in which the attorney receives a portion of the client's settlement or award as his payment, typically 1/3 of the total amount. If there is no recovery, the attorney does not get paid. The client never pays until the settlement or award is obtained (except perhaps to cover the filing costs for his claim). You can locate a lawyer in your area here.
You also may wish to consider paying a lawyer out of pocket to negotiate a severance agreement with your employer in which you waive your right to bring an FMLA retaliation claim in exchange for financial compensation. This is essentially an early kind of settlement. It may be appropriate here, and while attorneys can probably do this for you on contingency, it might make more sense to simply pay them out of pocket hourly since it could be that they can negotiate a settlement very quickly and you will save money by paying hourly rather than giving them a percentage of your recovery.
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