While the United States Constitution still applies, you are correct that the rules are different for civil cases than criminal cases. For instance, a person can be forced to self-incriminate during a civil case. Also the burden of proofs shifts from "beyond a reasonable doubt" to "more likely than not."
That said, I do think Jerry could try to appeal the decision of an agency [in this case the UIL] to federal district Court. Yes, you can do that per administrative code; basically, this appeals process embeds administrative actions within the framework of the US Constitution.
Although it would be a longshot, Jerry might prevail if he has nothing to hide. There is this little thing called Discovery, however, that happens in big boy court. With the government as the opposing party, those phone conversations might be heard by all of us. I am not 100% sure Jerry wants that.
Either way, I still do not like all the crap talk about Burkhart. He has kept us entertained on this site for years, instills hard work and a moral code in his players, and wins more than most coaches. If this is the "worst thing" Jerry has ever done, I think God has a pearly white uniform for Jerry Burkhart. I wish him the best as a relator and hope he comes back in 3 years.