From what I understand, the way canaries are supposed to work is, you keep publishing a document every day / week / whatever, that includes a phrase like "Today is 2016-09-02, and we have not received any national security letters or gag orders".
When you receive an NSL, you either publish a document including the phrase "Today is 2016-09-03." or else simply stop publishing that document and leave the one with the old date up, and let people draw their own conclusions.
Under this scheme, the court would have to compel speech on your part (force you to publish a new canary), but there's some precedent that might indicate that you can tell them to stuff it.
Courts can generally compel you to do things, especially things you don't want to do (otherwise the compelling bit is redundant). Also, from other people's writing on the subject, I gather that courts aren't all that keen on sophistry and schemes, especially those specifically invented to anticipate and circumvent a specific order from the court.
In the USA, courts generally can't compel you to speak. They do not have the power to force someone to utter or write words against their own free will. Forcing someone to continue publishing a statement saying that a warrant has not been received would be against understood Constitutional precedent.
Moxie Marlinspike: If it's illegal to advertise that you've received a court order of some kind, it's illegal to intentionally and knowingly take any action that has the effect of advertising the receipt of that order. A judge can't force you to do anything, but every lawyer I've spoken to has indicated that having a "canary" you remove or choose not to update would likely have the same legal consequences as simply posting something that explicitly says you've received something.
Bruce Schneier: Personally, I have never believed this trick would work. It relies on the fact that a prohibition against speaking doesn't prevent someone from not speaking. But courts generally aren't impressed by this sort of thing, and I can easily imagine a secret warrant that includes a prohibition against triggering the warrant canary. And for all I know, there are right now secret legal proceedings on this very issue.
When you receive an NSL, you either publish a document including the phrase "Today is 2016-09-03." or else simply stop publishing that document and leave the one with the old date up, and let people draw their own conclusions.
Under this scheme, the court would have to compel speech on your part (force you to publish a new canary), but there's some precedent that might indicate that you can tell them to stuff it.