At his stage, there is probably no public record from which the information can be accessed by a background search firm. The EDD does not make such information available to third parties. However, if the EDD does get to the point where it attempts to enforce the overpayment by going to court to obtain a judgment, it will become public record. If your husband loses his appeal and exhausts all his options, he should consider working out a payment plan with the EDD, to avoid a judgment and wage garnishment.
They say you get what you pay for, and this response is free, so take it for what it is worth. This is my opinion based on very limited information. My opinion should not be taken as legal advice. For true advice, we would require a confidential consultation where I would ask you questions and get your complete story. This is a public forum, so remember, nothing here is confidential. Nor am I your attorney. I do not know who you are and you have not hired me to provide any legal service. To do so would require us to meet and sign written retainer agreement. My responses are intended for general information only.