When people are hired into a job it is with the generally explicit expectation of certain job duties listed in the job description. This forms a very basic contract as to what they're hired to do. While they can be fired for not doing something else that's suddenly assigned to them, at least in "at will" states, they would have the right to collect unemployment for such a firing, as the new requirement is explicitly not their job! In other words, the firing would not be for (legal) cause.

But sure, many people do take PR jobs, where the general job duties include making the company (and by extension its officers) look good. Such a person could be fired for cause if they didn't want to make posts about how great the CEO is, and would generally be ineligible for unemployment.