The hearing is the place. If you have a diagnosis from a doctor, you can submit it. If you had tests or cultures done, you can submit the results. You can submit copies of your prescription labels. You can submit printouts from the internet for the type of condition for which the drug would be prescribed. You can submit photos from work showing water damage. If there was visible mold, you can show that too. You can use emails between you and your employer. You can printout phone records to show that you made a call to your employer on a certain date and time. Even though it won't prove the contents of a call, it will stop an employer from being able to say, "she never called me." You can also testify as to what you did; however, it always best if you can back it up. You can bring witnesses, but in practice, it's rare that they ever show up no matter how much they tell you beforehand that they have your back. If there was some a ticket/violation/report issued and it's available as part of a public record online, you can submit that.
As the quitter, the burden is on you. These cases can be lost because of just one tiny thing a claimant could have done differently, and this is pretty much your only chance to do it right.
You don't really know that. I had the same judge doing my board of review by himself and then part of a three judge panel. It was the SAME record that was being reviewed. In the first decision when he probably thought he could get away with it he says, "the claimant quit her job with the Employer because the Employer ceased to provide health insurance benefits. While certain fringe benefits may be considered wages, we (he really meant I) that health insurance paid by an Employer is specifically excluded."
Then with his peers, he says in a dissenting opinion, "I am in complete agreement with the majority on most of the decision. However, the majority is basing its decision on information contained within a more than 40 year-old UIPL (not true because it was reissued and strengthed in 1998 and was only 12 years-old at the time of my quit). Therefore, I would remand this matter to the Department on both issues." Which means instead of outright denying me like in his first decision, I'd have at least gotten to start all over again which wouldn't have been a total loss.
It's because of stuff like this, that I don't hold the UI people in high regard. Depending on who you get, the day of the week, if it's near lunch time, or who might be looking over their shoulder, you can get very different decisions with the exact same FACTS.

