Sounds to me like you were fired for walking off the job, which they have every right to do. You can try to appeal on that basis (though I'm not sure it'll work - walking off the job is also most often a disqualifier.
But a case could be made for your quitting, too. It's a matter of perspective (not of law).
There's no guaranteed way to find out what they might tell prospective employers. BTW, if it is their honest opinion that you were fired for job abandonment, they may legally say so (EVEN IF the unemployment office disagrees). You can ask a trusted friend to call posing as an employer or your can hire a reference checking service. You may not find out.

