Yes, they can fire you for personality differences, especially in "at-will" states.
First, let's go over what it means to be employed in an "at-will" state. To be employed in one of these states means that your employer can fire you for literally any reason outside of something that would be considered discrimination.
For instance, your employer can't say "I'm firing you because you're a woman and I just don't think you can do this job as well as a man." That's considered discrimination about your gender and is illegal.
What your employer can do in an at-will state is say, "I'm firing you because your attitude about this project isn't working for the team." On paper, the reason for being fired will probably be "personality differences."
In our situation, my husband was fired directly after turning down a promotion because it carried a lot of extra responsibility with an only ten-cent raise. On paper, his reason for firing was "personality differences." So, as you can see, this reason for firing can be used as a blanket reason for just about anything.