The former foreign secretary - who walked out of the Cabinet days after signing up to the package at the Prime Minister’s country residence - said Mrs May’s blueprint would leave the UK in “vassalage, satrapy, colony status” to the EU.
His comments came as website Buzzfeed reported that former Trump adviser Steve Bannon made contact with Mr Johnson during a recent trip to London.
A spokesman for Mr Johnson refused to comment on the report.
Mr Bannon, who is reportedly planning to set up an office in Brussels to bring Donald Trump’s brand of right-wing populism to Europe, repeatedly praised Mr Johnson during interviews in the UK last week.
But he also talked up jailed far-right activist Tommy Robinson as “the backbone of this country” in a bust-up with a reporter on LBC radio.
In his first public comment about Brexit since his resignation speech to the House of Commons last week, Mr Johnson said Mrs May’s plan for free trade in goods under a common rulebook with the EU “can’t and won’t work”.
Writing in The Spectator magazine, he said: “You can’t leave an organisation and still be bound by its rules. But that is what the Chequers white paper means.
“It is vassalage, satrapy, colony status for the UK. For the first time in a thousand years our laws will be made overseas, enforced by a foreign court.
“It can’t and won’t work. Chuck Chequers.”
Mr Johnson also defended the decision, made jointly with Home Secretary Sajid Javid, not to seek assurances from the US that two British jihadis would not face the death penalty.
He said Britain would have been happy to assist in the use of a drone to kill Alexanda Kotey and El Shafee Elsheikh if they had been located while still operating as Islamic State fighters.
“Would the British state, in these circumstances, have connived in straightforward extrajudicial killing? Too damn right we would,” wrote the former foreign secretary.
He pointed out that a drone strike was used to kill Mohammed Emwazi, another suspected member of the so-called “Beatles” cell responsible for the torture and murder of Western hostages.
“Of course we legally justify these drone strike assassinations as preventative: to stop future acts of terror in Syria. But that scarcely masks the reality that killing them is also retributive - payback for the filmed executions of innocent people,” said Mr Johnson.
“So why do we support these extrajudicial killings, with no due process, and panic at what might happen in American court?”
He warned that if they were not extradited for trial in the US, Kotey and Elsheikh could be “simply set loose, like so many other jihadis, to roam the streets of London again”.