I argued before that there can’t be a duty to speak, except in certain very specific cases involving a moral urgency. Hence, if you’re free to speak you’re also free to shut up. The freedom to shut up, although not recognized as a human right, can be important for the protection of other rights, either the rights of the person deciding to remain silent, or the rights of others.
For example, you have a right not to incriminate yourself (in the U.S., this right is translated into the Miranda rights and the Fifth Amendment). You may also want to remain silent because you want to choose your audience carefully. Some of your speech has to remain private, and your right to privacy would therefore be violated if you can’t remain silent in certain settings.
Or you may decide to remain silent in order to protect the rights of others: for instance, you may decide that certain words at a certain time and place would risk inciting others to commit crimes. Or perhaps your words may make it easier for others to commit crimes (take the case of a murderer asking you where he can find his intended victim).
However, the right to shut up is not just relevant in cases in which it can protect the speaker or others against rights violations. For instance, someone may refuse to pledge allegiance to the flag, take an oath on the bible, divulge his or her religious beliefs etc.