No. I'll take this the exact opposite pole on this one. No you do not have to accept your old job back. When you were terminated, by their choice, the relationship was severed.
After they have gone through the initial appeals process, and possibly the appeals hearing, and enough time has passed, they cannot reopen your approval and protest it any further. What they may do is try to question your on-going eligibility or whether you have refused suitable offers of work. The thing is, they, the separating employer is not going to be considered suitable work any more since they fired you. They didn't lay you off with potential for recall, they fired you. They, by their choice, severed the employee/employer relationship between you. Now, of course they are being penalized for this termination, since you are drawing benefits from their account. So they have quite a vested interest in getting you back into the labor force and off the unemployment benefits.
And they certainly can report to the system that they offered you your old job back. They certainly will do so. So be sure that you also report this offer of work to your unemployment system and get them to do a decision on whether or not it was a legitimate offer of work. You don't have a choice on this matter. They have probably already reported this offer. You must report it to the state as well. You will not get away with not doing so.
But when you report it to the UI system, you will by my experience and judgment ( and PLEASE, come back on here and tell us if this is not the case, since we seem to have some disagreement ) not be going to be required to take the job back. Be sure you point out to the unemployment system that this is the terminating employer, not just anybody else offering you a job. You don't want them to miss that part, and sometimes they aren't very good at noticing things.
