Absent a contract that states otherwise, you are an at-will employee. This means that you can be fired for any reason except those reasons prohibited by law. Reasons prohibited by law include things like your race, gender, age (but only if you're over 40), disability and/or religion. Nothing in your post indicates that if your boss was to fire you, it would be for one or more of these reasons. Under at-will, it is perfectly legal to fire someone just because.

It is perfectly legal for employers to give their employees assignments that differ from what they were hired to do. Your boss did not do anything remotely illegal by having you pack her things.

If you quit, you won't be entitled to UI benefits. UI only pays resigners if they can show they've been subjected to discrimination based on the reasons listed above. If you want to be entitled to UI benefits, you need to let them fire you.