I am the TN unemployment person. I have one question for you. Did you take your child to the doctor? Do you have a medical excuse for that last absence? If so, produce it. If you have to, go to his physician and get them to give you a medical excuse stating that you were at their office on xxxxxxx (that particular last absence) day, seeking medical treatment for your child. If you have this, you'll very likely to prevail in the hearing. If you don't, pretty much I'd say you won't.
We are assuming this is the hearing, after the initial decision, correct? You've already been told via letter that you have been denied, and told that you have so many days to file an appeal, and you have filed such an appeal, and are now scheduled for a telephone or in person hearing at which both you and the employer are present, correct?
If this is just a "fact finding" hearing, held before the first decision is made, you'll still have about the same chances, because you either did or did not take your child to the doctor or medical clinic on that day or not. There's not much else you can do at this point other than get that medical proof if you can get it.
They look at it that you cannot avoid being so sick you need medical treatment, or you cannot avoid having a child get so ill that you need to get them medical treatment. But even if the child was quite ill, if you didn't take them to the doctor for whatever reason, you can still be considered to have committed plain old misconduct if you are absent the number of times they consider excessive in their policy. Did you know about the policy, did you know that one more absence would probably lead to your termination before the day you had to call out due to your child's illness? That issue will come up.
The thing about whether you qualify for FMLA is another issue and has nothing to do with unemployment. We're talking about it because you mentioned "wrongful termination." This term has a very specific meaning, and is not much related to whether or not you are able to get approved for unemployment insurance. If you are approved for unemployment, it costs the employer money. Therefore they don't generally want to have you receive unemployment insurance after they terminate you.
To keep that from happening, they have to show you were terminated for a valid work related misconduct reason. Absenteeism is a very easy thing for them to justify, as they probably have clearly stated policies concerning attendance. Violate them without a really unavoidable reason, such as a medical emergency and they very likely can keep you from being approved for unemployment. If you can show that your last absence was unavoidable, in that your child was so ill they needed medical treatment, that is generally considered good reason for the absence, even though you knew you were likely to be terminated for it.
It is perfectly legal in MOST cases for them to terminate you due to absences, regardless of excuse, even with a terribly ill child (unless FMLA comes into play) but in general, the unemployment system says you can't avoid being sick or having your child get sick, if you can prove that it was a serious enough situation to require a medical treatment.