Knowing you were fired doesn't do anything to this claim. Public embarrassment isn't actually illegal, if the embarrassing fact is true. So, the fact that people knew you were fired and this was embarrassing simply is not a basis for a claim.
If you have lost some opportunities because of this sign, then yes you can sue them, but it's your legal obligation to prove those damages. For a claim to be a financially successful claim, you have the burden of proving substantial loss due to this issue. Furthermore, if you never took the steps to try and mitigate your damages, by contacting them and telling them to remove it, that could actually get your claim thrown out of court.
You must take the first step of contacting the company to tell them to remove it. If they choose not to remove it, then it might be worth seeking an injunction to have them remove it. The court could also issue a small award to you and pay your attorney's fees.
However, this is just not the sort of situation that lends itself to a lucrative lawsuit. Yes, you have a technical lawsuit here (after you ask them to take it down and they don't), but there is the financial reality that you have to keep in mind. Not all legal claims are worth filing, because the cost outstrips the potential return.