The First Amendment only comes into play when there's a state actor doing something.

The First Amendment argument is that the enforcement of an NDA under these circumstances would be state action.

A confession of judgment clause can, under some circumstances, be a deprivation of due process, even though the court would only be enforcing a private contract. I think that, in general, enforcement of a contract which deprives a party of a fundamental right which cannot be contracted away constitutes state action violating those rights.

Other than First Amendment I would look to arguments like changed circumstances, unconscionability... but I haven't put any thought into those just yet.