Re: You have (the right) to remain silent
Actually this is pretty much on the money. The first amendment protects speech from the government, but unless it puts you in a protected class (and even then usually within a narrow definition) it doesn't protect you from actions by private entities in response to what you say.
In this case they're arguing his comments violate his employment contract at the company by going against the ethics standards, which if true would make it entirely legal for the company to take action against him (either internal disciplinary action or legal action if that isn't possible for some reason)