Actually, if we're looking at the bill of rights, this might be more of a 1st amendment issue. This is from EFF's analysis last year of a previous attempt by DoJ to use the same All Writs law to compel Apple to break into a phone.
https://www.eff.org/deeplinks/2015/10/judge-doj-not-all-writs
https://www.eff.org/deeplinks/2015/10/judge-doj-not-all-writs
I have not yet had a chance to read Apple's legal response to the latest order, but I'm interested to find out if they used this argument.... the All Writs Act is not a backdoor to bypass other laws. The government cannot impose an unreasonable burden on Apple, and it cannot violate the Constitution.
...
But if Apple in fact has this capacity, or if the government instead tried to require it to prospectively reengineer the operating system on an unlocked device, All Writs is not the means to do so. Reengineering iOS and breaking any number of Apple’s promises to its customers is the definition of an unreasonable burden. As the Ninth Circuit put it in a case interpreting technical assistance in a different context, private companies' obligations to assist the government have “not extended to circumstances in which there is a complete disruption of a service they offer to a customer as part of their business.” What’s more, such an order would be unconstitutional. Code is speech, and forcing Apple to push backdoored updates would constitute “compelled speech” in violation of the First Amendment.