Former education secretary Justine Greening said the plans to follow European Union rules on trade in goods without being able to influence them was "the worst of both worlds".
"These nonsenses of threatening general elections and votes of confidence in the Prime Minister and as I actually said to the deputy chief whip "bring it on" because I shall be the first in the queue to give my vote of full confidence in the Prime Minister", she said.
Liberal Democrat MP Layla Moran, a champion of the anti-Brexit campaign group Best For Britain, said Mr Johnson was "playing a stupid game".
May said she was sticking with her plans. She said it would safeguard vital jobs in the aviation industry and keep Britain's tradition as a nation in the forefront of the aviation industry.
On the customs issue, May suggested a "facilitated customs partnership" in the meeting in her country retreat of Chequers which is ambiguous and according to the hardliners, a redundant deal that would not allow the United Kingdom to strike any useful deals with the countries outside the European Union, thus reducing it to a state that does not have a say in the European Union matters but continues to be governed by European Union rules.
Resignation statements like Mr Johnson's have previously been used by former ministers to inflict a departing blow on prime ministers with whom they have clashed.
But the scale of opposition to Mrs May's plan was made clear yesterday as MPs from both sides of the party lined up to criticise her plans with just months remaining before the country is due to leave the European Union on March 29, 2019.
Britain's Prime Minister, Theresa May, leaves the BBC after appearing on the Andrew Marr Show, in central London, Britain July 15, 2018.More news: British Open is his best chance of winning another major, Woods says
He told Today: "But our energies are going into negotiating a positive way forward with out European counterparts". I'm afraid the Prime Minister doesn't see that. I am just saying I think he would be a great prime minister. "I have come to the conclusion that.it is not what people voted for".
Meanwhile pro-EU diehards are also dissatisfied with her compromise position unveiled last week, and are plotting their own moves, hoping to make ties much stronger.
He described Ms Greening's call for a second referendum as "a little ill-thought out", saying: "If we wanted to extend the uncertainty for another long period this is one way of doing it".
Tomorrow, they vote on the Trade Bill - the one May's government is most anxious about.
He told Today: "The amendments are to a Bill that is created to prepare for the world after Brexit, to be able to establish new customs regime that will be necessary".
"He's Back", the paper said on its front page today.
The Chequers plan was "the only way to enable us to get a deal with no friction at the EU/UK border, to resolve the Northern Ireland border issue and to ensure that we can strike free trade deals around the world", said the spokesman.