You didn't involuntarily quit. You quit. You may have been tricked, but you quit. If that letter of resignation is at your hearing, you're sunk. Had it said, "You are telling me to quit or I will be fired. Here's my letter of resignation, tell me when you want me to leave." You might have had a chance, but that schedule and child care, thing is a real problem. You kind of have to live with it even though it's not what really happened. You either lied in the letter or you'll be lying at the hearing. Your credibility is shot.
As to the hearing, UI is about your LAST job. This one that you quit. What does your denial letter say? If you're lucky, it says that you have an able and available issue. In that case, you shouldn't have appealed.
Also, getting help the day before doesn't really give you much of a chance to get the help you need. At a minimum, get to the hearing early. See if the letter of resignation is in the hearing file. If not, try the "they said they would fire me, but give me a good reference if I quit instead." You might be believed, and it's what really happened so you say. Pray your employer doesn't show up or brings that letter at the last minute.
Next time it's quit or be fired, always go for the firing, and teach your kids this. This happens way too much, and it's been nearly 80 years that the UI system has been in existence, and yet people still don't know this.

