The Brexit secretary, Dominic Raab, has accused EU leaders of pulling “the handbrake” on negotiations at the Salzburg summit, saying the UK was offering a deal no other country would sign up to.

Raab said the UK would “hold our nerve” after a humiliating summit where EU leaders rejected the UK’s Brexit proposals. He said the UK had been “rebuffed on our plans without any coherent explanation as to why”. He added: “There’s these sort of rather dogmatic pleas to the single market unity but our plans were very carefully crafted around that.”

He told the BBC’s Politics Live: “We’ve revved up the motor of these negotiations, I’ve been out there a lot more frequently to get motoring, to make progress and the EU have just yanked up the handbrake. And for the negotiations to go forward they’re going to have to take their hand off the handbrake.”

Raab said the UK had been negotiating pragmatically, rejecting EU criticism that the deal threatened the integrity of the single market. “We’ve answered lots of questions, we’ve gone through the details. And I think some of the answers coming back have been dry legalism or dogmatic, computer says no,” he said.

“I say, ‘Do you have any other third countries, coming to you for trade talks, and offering to sign up, very controversially at home, to a common rulebook on goods and agrifood?’ They don’t.”

Dominic Raab on Friday. Photograph: Kirsty O'Connor/PA

Cabinet ministers and loyalists to Theresa May sought to defend her beleaguered Brexit strategy, insisting her Chequers proposals were still workable even after their rejection by EU leaders at Thursday’s summit.

It was up to the EU to “engage with what’s on the table” in the form of the Chequers deal, said the housing secretary, James Brokenshire, as the former minister Stephen Crabb urged May to “stick to her guns”.

“The easy thing for the prime minister to do would be to go with one of those two options that are being proffered,” said Brokenshire. “EEA, which does not deliver on what the people voted for in the EU referendum on freedom of movement, or Canada, which would effectively break up the UK.”

He was speaking on BBC Radio 4’s Today programme after the transport secretary, Chris Grayling, hit back for May on Thursday night, declaring there were no changes to the Chequers plan on the table and the EU’s demands on Northern Ireland were impossible for the UK to accept.

“The PM has set out red lines that this country is not going to stay in the single market, we’re not going to stay in the customs union – I agree with her on those, that’s the government’s position,” he said.

The attempted circling of wagons by Tory MPs in support of the prime minister – just over a week before the Conservative party conference – comes after EU leaders led by Donald Tusk and Emmanuel Macron rejected her Chequers plan as it stood, prompting Conservative hard Brexiters to demand she abandon it.

May was also set an October deadline for a solution on the Irish border issue hours after informing Leo Varadkar, the Irish taoiseach, in a private breakfast meeting that she felt it would be impossible to come to a compromise within such a timescale. A clearly nervous and angry May told reporters on Thursday that EU leaders were engaged in negotiating tactics designed to throw her off course.

Crabb said on Friday that the prime minister must not panic over Brexit negotiations and insisted: “There’s still life left in this. The prime minister can’t back down.”

“The first rule is: don’t panic,” the Pembrokeshire MP said. “One of the outcomes the EU leaders wanted from yesterday was for Britain to go away, push the panic button and rethink, but the prime minister needs to stick to her guns … the Chequers proposal is not perfect but broadly represents the kind of compromising package that protects Britain’s industrial base, that protects agriculture and represents a positive position to take into the negotiations.”

However, Crabb went on to say that the way in which EU leaders had sought to “belittle” May in Salzburg had taken many in the Conservative party by surprise and had pushed people like him to a position where they felt “the quicker we’re out of this circus, the better”.