Trump probably can't gag Bannon and 'Fire and Fury' author, say legal experts
Kinda the first rule of contract law is that a person cannot sign away their legal or Constitutional rights in a civil agreement. As the Declaration of Independence pointed out, your rights are inalienable and endowed by your Creator, and cannot be abrogated by a sleazy real estate salesman, even if he later becomes President.
One cannot create a non-defamation employment clause that prevents a person from testifying about his knowledge of the commission of a crime. In fact, legal scholars say it would be a crime for both parties - the person who agreed to take compensation for their silence, as the well as the person who offered compensation for silence. This much is certain: Bannon is free to report and testify about his knowledge of criminal activities.
Other Bannon statements may or may not be violations of his NDA. The First Amendment issue is tricky in this case. The amendment forbids a law stifling free speech, but does not prevent private individuals and institutions from doing so. This is why corporations can force you to sign a non-disclosure agreement. The tricky part of the issue then becomes this: a presidential candidate is a quasi-governmental position. When Bannon was working for Candidate Trump, or President-elect Trump, was he employed by the government or by a private individual? That is a matter that would have to be decided in court.
On the other hand, Bannon is free to dish any dirt he wants on President Trump, unless it compromises national security or legally protected information. (He can't reveal troop movements or Trump's medical info, for example). Once Trump took that oath of office, he absolutely lost his right to stifle criticism with an NDA.
I find it unlikely that Trump would follow up on any of these defamation suits, since he would have to prove the statements were both false and malicious. I hope he does follow up. Talk about entertainment! If he pursued that course, the lawyers for Bannon or the publisher would have to subpoena the President to defend in sworn testimony that the offending statements were false. They would absolutely swamp the President of the United States with depositions and written interrogatories. This is not how we, the people, want our President passing the time, and this is not how Trump wants to spend his time. For one thing it would really cut into his golf game. More important than that, by testifying under oath, Trump would either have to air all his dirty linen or lie under oath, neither of which is a good option. (Especially if there is any truth to Wolff's claims to have taped conversations.) You can't just deflect or bullshit or say "believe me" in sworn testimony. Those lawyers will keep following up and re-phrasing the questions until they get a definite answer. Bill Clinton is a gifted liar and a brilliant guy, and he got caught in this trap. Imagine how much worse things would turn out for Trump and his monumental ego.