I've had a similar situation myself (Albeit I'm from the UK). It was my first part time job, at the local village shop, not a busy place, a few regulars and some passersby really.
Anyway, I was working for a particularly horrible boss. Constantly changing his mind on how to do things, refusing to accept the blame for things and just pinning all mistakes on other people, or me in particular as I was admin-trained.
Anyway, one day I was working on the till at the end of a 4 hour shift, which finished 12:00 noon. The boss came to let me know I could go, so I took the opportunity to tell him I thought I'd made a mistake with the way I key'd in a non-barcoded item.
There was a queue of 3 customers who he had just started serving. He stopped, looked at me and just shouted non-stop for 3/4 minutes straight in front of the customers. Explaining that, although this was something he'd changed his mind on about 4 times in the last 3 months, I should be perfect and not make the odd mistake here and there (Which was easily correctable - The cutomer was my best friends mum who lives 3 doors away from me).
Anyway, I left the shop the go home and on the walk home was furious at how he had been treating me and had shown me up in front of the customers.
I was scheduled to work 12-6 the next day, but instead I text him that same night saying I wouldn't be going in for that shift or anymore (Along with reasoning) and that was that. He replied simply, "Good luck.. !".
To this day, although my brother works there and family regularly go in, I've not seen or made contact with him.
However - I do still include the experience on my CV, without mentioned how it ended/why.
I'm now in my 9th month as a software developer and before that had a bar-tending job - Neither employer found out about what happened or asked for a reference from the previous job.
Long story short; If you do it, include the experience on your CV/resume, but ideally give notice, unless in an exceptional circumstance. I certainly don't regret quitting with no notice, but it is generally best to just cover all basis - And with no contract, you don't have to give a set time period notice. A few days will be enough.