So, for starters, what is described in the article (and maybe even the post your responded to) is not really a termination without cause, it is a termination for cause.

If you have a policy you require of your employees("You must take a picture of yourself"), and you terminate the employee for not doing it, that is not a without-cause termination. That is a termination for-cause - the cause being the violation of the policy.

Termination without cause would basically be "we are downsizing, you are no longer needed".

The rules about "without-cause" are pretty consistent state to state. However, the rules on "with cause" vary state to state. In a lot of states, you can't terminate someone for violating a policy if they didn't violate it. These policies are a form of "implied contract".

In other states, the employer can act completely and totally arbitrarily, as long as it is not acting illegally.

This is actually the minority of states.

See https://en.wikipedia.org/wiki/At-will_employment