How much would this impact my career afterwards when I look for a job? Do companies care if the university I go to is unaccredited (ABET) or not? How big of a disadvantage is this? I was doing some research online and was unsure where companies stood on this.
I think it depends on the college, and on the employer, so there's no one good answer on that.
I would highly recommend against it though:* You will be disqualified from Federal government jobs requiring a degree. They don't recognize a degree from non-accredited institutions.
* Very few accredited universities will accept credits from non-accredited schools. If you want or need to switch schools partway through your undergrad, you may have to repeat courses. I know this one from experience.
* If you decide at any point later in life that you want to do a master's degree, same deal applies. A lot of schools won't recognize the BA/BS from your unaccredited school. And you sure as heck don't want to repeat your entire undergrad degree in order to do a Master's. Watch out for for-profits like DeVry on this one, too.
I guess the question is: what is leading you to a non-accredited school? If it's a religious college, you might have some leeway with employers, especially if you are sharp and get good at what you do. But in general, I'd recommend avoiding them. And if it's a for-profit school like University of Phoenix. H-to-the-no on that.
There are accredited online schools these days worth looking into that are relatively inexpensive.
For DeVry and University of Phoenix, their Bachelors in Business has the same accrediting body (ACBSP) as Texas A&M. http://www.acbsp.org/?page=membership_list
I'm not saying that they have great reps, but we shouldn't give the false impression that they're "unaccredited".
I couldn't say how many of their general ed classes are, or are not, fully transferable. But my classes from fully-accredited West Valley College in Saratoga didn't all transfer to San Jose State University, either.