David Davis, the former Brexit secretary, has insisted leaving the European Union without a deal is not a frightening prospect but admitted there would be “hiccups” in the first year.

Speaking on BBC Radio 4’s Today programme he argued a no-deal exit would be better than plans Theresa May is trying to get her cabinet to back.

He said: “I don’t think no deal is as frightening as people think … We would leave and there would be some hiccups in the first year, no doubt. But we would have all the rights and controls over our own destiny [and] our own future.”

He dismissed fears about shortages of food and medical supplies and queues at the channel ports as “proven nonsense” and “scare stories”.

He added: “We are a big country, we can look after ourselves and we will look after ourselves very well indeed once we are properly out of the European Union.”

“The best deal is a free trade deal, but don’t be terrified of a World Trade Organisation deal, after all most of the rest of the world trade successfully with the EU, in some arguments more successfully than we do, under the WTO rules.”

The Brexiter MP also heaped pressure on May to publish full legal advice on the vexed issue of the customs backstop to avoid a hard border in Ireland.

He warned that if the prime minister continued to withhold the advice, parliament would be more likely to vote against her deal.

Davis said: “If the deal fell in parliament, which at the moment is looking like a probability certainly if they don’t release [the legal advice], then it would have to go back to the union once more. Both sides would be staring at no deal.”

Davis speculated that May was worried about publishing the full advice because it would expose flaws in the government’s arguments over the backstop mechanism.

He said: “The only thing I can think of is that they feel there are weaknesses in the public argument about what they are presenting.”

He added: “Are we going to have to wait until the Irish government says it is OK to leave? If so, that is not acceptable. Are we going to have to wait until its convenient to the commission? If so, it is not acceptable. And I suspect that they have not pinned down any of these issues and they need to be pinned down before parliament votes.”

Davis said the cabinet and parliament needed to see the full advice before deciding on whether to back May’s proposal.

“What we must see is the complete legal advice, not a summary, not something that glides and glosses over all of the difficult issues, but the actual legal advice,” he said.