If I do not commit to it, would I be eligible for unemployment? I am not able to meet the schedule requirements -- would this be considered them firing me, or me resigning?
You've got a decent shot at getting UI, but the big factors are just how big is this change in hours, what is your occupation, and what's a prevailing schedule. It's going to get treated as a quit, but make your life easier and tell DES that you were fired, and not another word. You need to see what your employer has to say so that you present your appeal correctly if it goes that far. Make absolutely sure that you have proof (preferably with an email) that you tried to get your employer to leave your schedule alone or that you suggested an alternate schedule that you could live with.
Then let's reference the Arizona Administrative Code at http://apps.azsos.gov/public_services/Title_06/6-03.pdf
You can work with R6-3-50155 Domestic Circumstances B. With good cause in connection with the work if 2. The hours of work or place of employer were changed
You also might be able to work with something in R6-3-50450.
Just know that while you might get a nondisqualifying separation, it's going to raise able and available issues so you still might end up getting denied benefits if your schedule is so rigid that it's determined that you aren't available to a substantial field of employment.
In all probability, this is going to get treated as a quit, and you have to prove everything so make sure that before you leave work you have proof of:
What your schedule currently is
What the schedule is going to be
That you tried to get your employer to work with you.
Without proof of these items, at an appeal hearing your employer can say, "we never said we were changing her schedule," "that wasn't her schedule," or "this is the first we're hearing of it," and you will lose.