No. If the employer wants you to work in absolute silence, it can make a rule and enforce it. Your choice is to find another job if you do not like it. That is the nature of the at will employment relationship.
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A manager can tell his or her employees to do just about anything that does not specifically constitute a violation of law or public policy. While a rule against talking is certainly a bit harsh, it is not illegal nor does it run afoul of any public policy. Therefore, it's legal, and your options are to either do what your employer is asking or find other employment. I'm sorry.
This answer is a general interpretation of the law and is not fact specific to your case. Likewise it does not create an attorney-client relationship. You should seek an attorney for a review of your specific facts and documents.
My collegues are correct according to the facts you've given us. There are limits and exceptions as always, such as under the NLRA, but from the facts you've given it doesn't appear that those limits and exceptions apply.