I would argue that stopping to publish these statements is equivalent to announcing that some warrant has been served. So if the latter is a criminal offense, why should the former be legal?
There are supreme court precedents finding that the first amendment prohibits the government from compelling speech, which is the legal theory for why warrant canaries are not equivalent to directly disclosing the NSL's existence: https://en.wikipedia.org/wiki/Warrant_canary
Nobody can say, “you have this on your web site saying you have never received a warrant; here’s a warrant, and by the way, you must also continue saying (now lying) that you have never received a warrant.”
The government can tell you to /stop/ posting a daily or weekly notice. It can say “here’s a warrant; you may not tell anyone you have received this.” In that case, you would be lying to say you had /not/ received it, and discussing whether you have received it or not received it at all can be prohibited. In which case, you would be obligated to remove or cease updating any statements related to a warrant.
*I say "in theory" because I understand that this was never tested in court and some big name law professors did not agree that this would stand up in a court of law. A judge would look at the spirit of the law that bans the warrant disclosure and consider that the entire setup of the warrant canary is aimed at breaking it. I tend to agree with this interpretation because any reasonable person would consider the whole design is aimed at breaking that law.
The point of gag orders is to keep a criminal from destroying evidence while an investigation is underway. Having an individual warrant canary for every customer would defeat this purpose. A general canary doesn’t appear to.
It would be a sad day for US justice if a judge ruled that every action was within the law but, because a reasonable person would conclude that the actions in aggregate were intended to circumvent a law, the defendant is guilty.
I would not assume anything about warrant canaries, and would expect the FBI et al to get the judge who gave them a gag order to also issue a warrant canary falsification order at the same time. Most organizations would not have the werewithal to resist, or the fortitude of a Judy Miller or James Risen and their willingness to go to jail rather than reveal their sources (yes, I know Judy Miller is a warmongering sorry excuse for a journalist, but her stand in this respect is admirable).
Also gag orders aren’t perpetual. If there were orders compelling businesses to lie, past examples would have come to light the same way we know gag orders exist.
If the Feds really want to nail you -- to teach you a lesson and/or make an example out of you -- you're nailed.
"Democracy Versus The National Security State" by Marcus Raskin, 1976
The Supreme Court has repeatedly ruled on compelled speech [1], most colourfully on the pledge of allegiance [2]. It’s the theory Apple ran with in the San Bernardino case to avoid creating a decryption key [3]. A theory that was so successful the FBI pulled their case rather than risk a precedent for encryption.
[1] https://en.m.wikipedia.org/wiki/Compelled_speech
[2] https://en.m.wikipedia.org/wiki/West_Virginia_State_Board_of...
[3] https://en.m.wikipedia.org/wiki/FBI–Apple_encryption_dispute
But the issue hasnt been settled in the courts. Twitter v Barr (2014) attempted to settle the issue, but the case was dismissed in 2020.
Edit: https://www.courtlistener.com/docket/4181259/twitter-inc-v-b...
More succinctly, the legal precedent would compel people to "break the law".
I'd argue such a conflicted legal outcome could only be resultant from unsound base principles.
If the government can compel you to hand over some information, well that's government. They can also injunct you not to tell people.
But, can they compel you to lie, publically ? Can they compel you to not publish such canaries - just in case they want to compel you later on?
And frankly, does it matter - is such a canary really just a form of social performance? If I don't want the US government to know something about me, I should stay the hell away from modern life, internet and so forth.
This is the whole contention with Microsoft Office 365 being hosted in the EU and the CLOUD act.