Under the plan, the United Kingdom would continue to follow EU regulations relating to goods and farm products but it would leave the bloc's formal customs union, which provides for the tariff-free movement of goods among member states.
In a much anticipated speech, Mr Johnson attacked the prime minister's approach but said she could still command support for her leadership.
May's political future has been uncertain since her "white paper" policy document on post-Brexit relations with the European Union triggered resignations from the Cabinet and anger within her Conservative Party.
"We have a duty of care", Barnier said.More news: U.S. dollar down on as Trump makes fresh threats vs China
In their final exchanges before the parliamentary summer recess, the Labour leader said she was neglecting the country's needs because she was caught up in Tory infighting.
We will only have a withdrawal agreement if the legal text is agreed on all issues.
"With serious issues yet to be resolved, not least at the border of Northern Ireland, the prospect of a catastrophic no-deal Brexit is becoming more concrete by the day". Senior Conservatives said afterwards that she had weathered a febrile few days at Westminster, and would survive to resume battle with Brexit hardliners after the summer recess.
In a letter to his constituents, he had said May's Chequers plan for Brexit was "unacceptable" and claimed failure to deliver the highly controversial Brexit could end with Jeremy Corbyn as prime minister. Some have gone so far to call for a "national unity government" made up of Conservatives and Labour MPs to deal with Brexit.
"When you make a proposal limited to agricultural products and goods, you have to ask what about services, what about capital, what about the free movement of people".More news: DRAGON BALL SUPER: BROLY Trailer Sets Up a Huge Fight
He was flanked by the hardline Eurosceptic Nadine Dorries, and accompanied by the former Brexit secretary David Davis, whose own resignation over the Chequers deal pre-empted Johnson's.
"I'm looking forward to with new energy, vigour and vim looking at the detail of all of this".
"We are quite happy that we have something we can work with, but it is not more than that, it is not the ten commandments, it is a negotiating paper put on the table", said the European Union official, adding that the European Union doesn't negotiate the white paper, but the exit deal. "That is why I'm a little bit nervous", Germany's European affairs minister Michael Roth said in Brussels on Friday.
May said the return of a hard border once Britain leaves the bloc would be "almost inconceivable", but dismissed the EU's current plan to avoid it as unacceptable.More news: Sens. Jeff Flake & Chris Coons Russia Resolution Blocked