Yes, it is actually an IRS thing, and they could lose their right to offer a dependent care account altogether if they illegally reimbursed you. Because this is a pre-tax benefit, it is VERY heavily regulated by the IRS and subject to audit. They do not have the option of reimbursing you even if they want to and even if you are willing to pay the taxes.
IF they offer a medical FSA AND IF they choose to call it an administrative error and transfer the funds to it AND IF you actually have bills incurred BEFORE your termination that would be reimbursable under a medical FSA, then you can get back some or all of it in the form of reimbursements. That is the ONLY way you can legally get any of the money back. You are NOT going to get it back as income under ANY circumstances. You will have to provide them with copies of the reimbursable expenses IF they are willing to do this for you at all.
However, as I said, since by your own admission this was your error, not theirs, they have no legal obligation to take responsibility for the mistake (which is what they would be doing by calling it an adminstrative error). And if they don't have a medical FSA, the whole thing is moot. Likewise it is moot if you do not have any reimbursable expenses under a medical FSA.