the only thing better about quitting than being fired is being fired doesn't look good for future employment
ideally you'd like to be able use former managers as references... and if they ask "Reason for leaving" you don't wanna write down "they saw pictures of areolas on my computer while I was browsing an gangster rap internet forum"
although I've heard a lot of managers won't even say shit about you, unless you're actually listing them as a reference maybe... but if you're just listing their name as a former supervisor on an application they'll just confirm whether or not you worked... not sure how true it is
but yeah, if you're gonna try to get unemployment just say you were unjustly fired because everyone there violated internet policy"
they're gonna get your side of the story, then they're gonna contact your employer, your employer will give his/her side of the story, and they'll make a decision
my dad owns his own small business, he's had some shitty ass employees that he fired, they filed for unemployment, the EDD office called him, he gave his side of the story and the employee got denied UI benefits..
good luck though man... it's worth a shot, but don't depend on it