Paraphrasing from a UI help pamphlet:

If an employer makes big changes in your job (duties, hours, pay, benefits)you may be eligible for UI. I think day care issues, however, are personal reasons and not job related and not "good cause" for quitting.

Did you try and work out the schedule issue? Do you have documentation? Witnesses? It seems that if the employer acted unreasonably, you may be able to make a convincing argument that the unknown work schedule was unreasonable to any employee - let alone someone with children.

If you quit in a huff, you could also argue you had no intention of quitting - you need to convince UI quitting was an emotional reaction to unreasonable work conditions. Did you try and get your job back? Maybe there are legitimate reasons why you didn't or couldn't? You can not win any appeal if you do not appeal. An appeal costs nothing. You don't need an UI attorney unless you have cash to pay for one. Find an experienced paralegal. Just be honest and get the documentation to the hearing officer before the hearing. If the company has a bad track record, get the documentation. You can win. Good Luck!