The Prime Minister's Office says May isn't abandoning her Brexit strategy, and will fight a no-confidence vote.
"The general direction of policy will leave us in at best a weak negotiating position, and possibly an inescapable one", Davis said in his resignation letter to May.
He said May's plan may be worse than leaving the European Union without any deal at all.
"The cost, complexity and bureaucracy created by crashing out of the customs union and adopting alternative arrangements is the last thing that our businesses need as we seek to grow", they said in an open letter reported in The Times.
"With her government in chaos, if she clings on, its clear shes more interested in hanging on for her own sake than serving the people of our country", Corbyn said.
Mr Bone was heckled as he added: "They said they were betrayed and they asked, 'Why do we go out each and every Saturday to support the Conservative Party and get MPs elected?'"
The UK would of course continue to play a strong role in shaping the global standards that underpin them, and Parliament would have oversight of the incorporation of these rules into the UK's legal order - with the ability to choose not to do so, recognizing that this would have consequences.
He criticised May's decision to maintain a "common rule book" with the European Union, mirroring the bloc's rules and regulations, saying it would hand "control of large swathes of our economy to the European Union and is certainly not returning control of our laws".
She said the commitment to end the free movement of people would be met, the jurisdiction of the European Court of Justice in the United Kingdom would be curtailed, and the U.K.no longer would send "vast sums of money" to the EU every year.
Among the biggest Brexit-supporting names in the government, Trade Secretary Liam Fox put his name to a newspaper article backing the plan, and Environment Secretary Michael Gove defended the agreement in a television interview.
His resignation seemed to spur others to follow suit, with a source saying that a junior minister in the same department had also quit, just two days after May had held a crisis meeting with ministers to overcome the deep divisions over Brexit.
Mrs May has consistently tried to go over the head of Commission chief negotiator Michel Barnier and straight to European Union leaders.
The outgoing minister suggested that Mays promise that Britain and its parliament would "take back control" from Brussels was hollow.
First Davis and then Johnson appear to have done so, and their resignations have come at a critical and highly sensitive time for May. It was a rare consensus, following talks at her Chequers country retreat about the way forward - a move meant to kick-start talks with the European Union that have been stalled for months.
He said Brexit was a "lose-lose game".
Merkel again said it was the European Commission, rather than member states, which was leading the Brexit negotiations, possibly dampening May's hope of winning support from Germany before the so-called Chequers away day.
But in an effort to win over critics, Tories were being invited to briefings about the plans, with the Prime Minister set to address Conservative MPs at a meeting in Parliament on Monday. Some lawmakers have already expressed their misgivings.
Jacob Rees-Mogg, leader of the influential European Research Group of pro-Brexit Conservative MPs, warned that a common rulebook could make "trade deals nearly impossible" if it meant regulations would have to apply to any goods coming into the UK.
"[Mr Davis'] departure raises more questions about what came out of that meeting".
May replied to his letter to say she did not agree "with your characterisation of the policy we agreed at cabinet on Friday".
Writing in the Mail on Sunday, he said Jacob Rees-Mogg was now the only credible challenger to Mrs May, after Brexiteer cabinet ministers failed to oppose the PM's plan. "It's more important to do what you have promised the British electorate than stick by lines in the sand".