Is it legal for a landlord to tell everyone about your private business?

Any person is free to talk to others about matters that concern them. There are laws and rules that limit what information may be disclosed publicly, but the limitations of this kind are usually related to who that person is and what his relationship is to the person that is the subject of the disclosure. For example, a professional person, such as a lawyer or a doctor are prohibited from disclosing anything. But a landlord is not in a special relationship with you. He is not subject to these restrictions. He may disclose information that is true. He may not tell others lies about you, if written that is "Libel", if spoken that is "Slander". If he told other tenants that you were behind in your rent, and its true, then you don't really have recourse. However, If he is telling people not to do business with you, then that is a different issue. If he is a customer and he is not satisfied with the service he is receiving, he may be vocal in sharing his opinion, that is not actionable. However, if he goes beyond stating his opinion, and is telling lies, and his actions go far beyond what could be merely telling his opinion, he might fall into another category of actions, Tortious Interference With A Business Relationship. Cases like these always depend upon the very specific facts. You should consult an attorney. An attorney may be able to send this person a Cease and Desist Letter warning of dire consequences if he continues.

Why can't he? You are a business and he is talking about business issues. Once it is out in public, it isn't private anymore.