Quick 2 cp here:
You don't need to have a BS or BA to work in video games. In general, you just need to be able to show the employer that you can do the work that they need done. This generally means that you need to have experience before you apply.
That may seem like an impossible obstacle, but it isn't. When I was trying to break into the industry, I attended a convention where Robin Walker was speaking. He gave me probably the only useful piece of advice that I've ever received: "If you want to start making video games, you need to start making video games". Basically, whatever it is you think you want to do, start doing it. If you want to be an artist, start modeling and texturing. If you want to be a designer, start making maps and missions for existing game englines. If you want to be an audio designer, start putting together effects, voiceovers and music. If you want to be a software engineer, start programming. There are a gazillion tutorials on the internet for all of these things, and forums that will provide you with answers you can't easily google. Build up a portfolio and a website, start showing off what you've done.
And once you're getting pretty comfortable with your craft, the way to really sell it is to make a mod or develop a small game from scratch with a team of people. If you are part of a team that makes something which other people enjoy and attracts attention, you have about 1000% higher chance of getting into the industry. It's also a good way to know if you'll actually like the development process or not; working late hours with a bunch of different people trying to find some crazy ass way to glue things together is a pretty common thread in development. There are definitely some big differences between professional development and amateur, but in the end the process is about the same thing; making something fun.
That said, let me backpeddle a bit; if you want to be a programmer, you will be severely limited by the merit of your Computer Science / Computer Engineering degree. It is also unlikely that you will quickly rise to an Art Director without a degree from a Fine Arts school ( and a great portfolio ). Having the academic background means a lot, but it isn't a total barrier for entry.
Also, a quick word about QA: it can be a rough way forward. It is appealing to many because it is probably the single easiest way to get "in" to the industry. There aren't many ( if any ) requirements around education, or specialized skillsets, and there's always a need for lots of testers. But getting into that designer position from QA is a long, long hard road. Getting peopel to view you as not "just a QA guy" can be difficult, especially at larger companies. You aren't going to get any real developer training on the job, although you do get a pretty good sense of the development cycle and what makes games work ( which is why some QA folks eventually move into Producer roles ).
If you really really want it, you can make it happen. But you will need to make sacrifices. You may need to move. You may need to learn a new skill on your own. You may need to give up parts of your life to make time for it. In the end though, it's worth a shot if you think it might be the only thing you want to do.
Hope this helps, and good luck with whatever you choose!
