Can an employer demand that their employees never mention their store or speak of any products carried by them?

Yes, and the company's position will be upheld, even in a liberal state such as CA. The employer has the right to control its image: this includes facebook posts by employees, even those as mundane as "I work at ________." The company would do better though to issue nondisclosure agreements and make the employees sign them.

Yes

They can easily enforce such a ban, just to prevent the usual whining about judgement calls.

while an worker is suspended they're many times removed from the place of work and instructed to no longer enter till disciplinary technique has taken place and the difficulty resolved.the only touch the worker could have is with their representative to talk their difficulty no longer yet another worker.it might desire to complicate an analyze whether it is got here across that different workers are discussing it with the guy who has been suspended, yet i won't be in a position to ascertain the place they might end you chatting with them as long as this is not approximately their difficulty

in what context........if you mean outside the store then they can ask...............on line yes they can etc etc.

Bit vague really to be able to help you

EDIT Yes totally they can...........the retail store I work for has put up new and revised guidelines by the time clock etc etc.......I think that people should not really even put where they work to be honest on places like FB.............everyone that knows me knows where I work............

An employer can demand anything they want, as long as it's not illegal. If you don't want to comply, you are always free to go find another job.

Of course they can. If you don't want to abide by this policy then nobody is forcing you to stay there.

Yes because you would be accused of espinoge.

of course.