Regarding background checks: at a US university, your educational records should be protected by FERPA and nobody should be able to access them without your permission.
However, you should probably assume that a potential employer would be able to learn the following things: You used to be enrolled in the PhD program, you're not enrolled now, and you didn't receive the degree. (Some of this would be "directory information" which FERPA does not protect; other parts might be otherwise publicly available, such as from old department web page listings of grad students.) So they can reasonably deduce that you either quit voluntarily or were kicked out ("terminated").
Some employers might also require, as part of the job application process, that you give them a copy of your official transcript from the university. If you are terminated from the program, the transcript will state this, and probably explain the reason ("didn't complete program requirements within time limit", "didn't make satisfactory progress", something like that).
If you don't think you will be able (or willing) to either defend before the deadline or negotiate more time, you might see about voluntarily withdrawing from the program. ("You can't fire me, I quit.") This might not look as bad on a transcript. After all, it's not uncommon for people to start graduate programs and then decide it isn't something they want to pursue, and you could explain it to an employer in those terms. ("I decided that instead of academic research, I wanted to work in industry, so that I could do work that was more practical / real-life / lucrative.")