The Volokh Conspiracy
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Lynyrd Skynyrd airplane crash movie blocked by judge, on breach-of-contract grounds
Eriq Gardner of the Hollywood Reporter has a summary, and you can read the opinion here; Pages 55-60 discuss (and reject) the First Amendment defense. The bottom line:
Cleopatra [Films] is prohibited from making its movie about Lynyrd Skynyrd when its partner [Artimus Pyle] substantively contributes to the project in a way that, in the past, he willingly bargained away the very right to do just that; in any other circumstance, Cleopatra would be as "free as a bird" to make and distribute its work. The First Amendment is not infringed by such private and voluntary decisions, and the Court has the authority to protect those previously bargained-for rights.
The Supreme Court has indeed held that damages can be awarded for breach-of-nondisclosure agreements, see Cohen v. Cowles Media Co. (1991); it hasn't discussed whether courts can also issue injunctions against speech that breaches a contract, but lower courts have generally concluded that such injunctions are allowed, see, e.g., Perricone v. Perricone (Conn. 2009).