Not a US-based company, because we have these nefarious things called National Security Letters. The penalty for knowingly breaking the gag order with the intent to interfere with an investigation is up to five years in prison.
https://en.wikipedia.org/wiki/National_security_letter
Do you know any business owners willing to go to prison for five years for one of their users?
The answer to the above is still no, however. I don't see how "I post everything I receive" could possibly get you out of "don't post this warrant".
Imagine you hire a number of people who happen to be willing to go to jail for their principles, who are strongly opposed to things like government secrecy, and put them in the line to handle a NSL. If they happen to violate a gag order and go to jail, well, you may have to fire them, but their contract is rather iron-clad and their severance quite impressive, more than adequate to take care of their family during a five year jail sentence...
There really is no prevailing over the legal tarpit - even if your action technically passes, they get you on intent or association. The only time logic really comes into play is writing post-hoc justification of the status quo. The more corrupt "laws" created, the easier such justification becomes.
(PS. given that authoritarianism is creeping in on many fronts, everybody who cares about these issues should be looking for ways to be subversive. embracing one front in an attempt to outrun the others is a losing strategy)
I can't imagine a business owner going to prison as you suggest.
(a) in the form of a valid court order hand-delivered to and served on your CEO at his home at 6am by two very grim federal agents who don't really like getting up this early, or
(b) if you're represented by counsel, it comes in the form of notification to your attorney by the court clerk that a court order has been issued (or perhaps notification from an AUSA if the clerk's office is slacking).
OpSec notwithstanding. If the government doesn't know your other email addresses, this obviously doesn't apply.
If everything that comes to every one of your inboxes is immediately laid bare for the world to see, they cannot circumvent the process and also cannot prove that you knowingly and intentionally violated the gag order. I don't know anyone who does this, however.
The Lavabit guy came pretty close, IIRC, and he's a small fry.
Do you really think the government would jail the wealthy executives of Google or Microsoft? They won't even prosecute Wall Street executives, and bankers are far less popular with the public than tech CEOs.
They can, they have and they will. Look up Joseph Nacchio.