Different states administer their own unemployment insurance programs, but there are some federal guidelines that are common to all. Due to the astronomical rate of of unemployment at the moment, the program is being stretched beyond capacity: we're paying unemployment compensation from funds that are essentially nonexistent. This is not a favorable environment for filing a claim of marginal validity.
In broad general terms, to be eligible for unemployment compensation you must have:
1.) been working
2.) become unemployed through no fault of your own
3.) be available to work
4.) be actively seeking and willing to accept any reasonable employment offer
In making "the just cause" argument, only #1 works in your favor, the others are going to be a problem for you. You may have needed an indefinite leave of absence, but you quit your job instead. If you can prove that you sought a leave of absence and your employer denied it, you may have an argument that is a little bit stronger (if you have proof
of the denial). You are also apparently not currently seeking employment which might disqualify you no matter what. You have a right to quit your job, but not necessarily a right to have the public subsidize your living expenses while caring for your family member. You have to be available to work
to qualify for unemployment benefits. The same goes for pregnancy and parental leave. If you're pregnant and able to work, yet quit your job, you will not receive unemployment benefits.
It's important to understand that the outcome of the hearing will be based on the express language of applicable state statutes
and not on sympathy or merely an argument that sounds reasonable. Unemployment eligibility in Arizona is established under ARS 23-771 (see link below). The only argument that I can really think of is that your are not "able" to work under ARS 23-771(3). You are physically capable of working, but you are situationally "not able" to work because your time is entirely monopolized by the need to tend to an ailing family member. Emphasize that:
1.) You did not want to quit the job you had.
2.) Your unemployment was the result of a Hobson's choice: you had to quit your job in order to alleviate a family hardship.
3.) The hardship you sought to alleviate was not self-inflicted.
4.) You did what any reasonable person would have done when faced with similar circumstances.
I'll be honest with you -- it's a weak argument. There are tens of thousands of caretakers who are doing what you're doing and they're not getting paid with public money either. If you lived in Germany, you might get some compensation. The U.S. is weak on family security issues, though, and Arizona is a "right to work" state, so social justice law is particularly feeble there. If there are any medical orders that were written for hospice care for your mother prior to you quitting your job, bring them to the hearing. Good luck. You've got a tough row to hoe.