The former foreign secretary said: "The fatal error was not to challenge the EU's position that the only way of avoiding a hard border on the island of Ireland - an objective we all share - is for Northern Ireland to have the same regulations for trade as Ireland and the rest of the EU".
"What we can not do is see the United Kingdom locked in via the backdoor to a customs union arrangement which would leave us in an indefinite limbo - that would not be leaving the EU", he said on Friday.
"We are working not just with the domestic preparations that we need to do, but with the Belgians, the Dutch and the French, to make sure not just passengers going, as you say, on holiday but also goods can continue to flow".
Mrs Foster said the PM should learn from Margaret Thatcher who "deeply regretted" being persuaded to sign the 1985 Anglo-Irish Agreement, which gave the Irish government a formal say in Northern Ireland affairs for the first time.
The Brexit Secretary said that a provision on a customs union which was not "finite" would fail to deliver the result of the 2016 referendum. "The rigid negotiating stance of the European Union, and refusal to offer any form of concession to the United Kingdom government has made the prospect of the two extreme ends of Brexit, these being a very soft "Brexit in name only" or a "no-deal" hard Brexit, more likely", read the report, which came out Monday.
Chancellor Philip Hammond became the first senior Government figure to suggest that the backstop - created to keep the border between Northern Ireland and the Republic open in the case that no broader EU/UK trade deal is finalised - will "probably" have to come into effect for a period.
Sri Lanka v England: second ODI
England , in the first ODI against Sri Lanka , breezed to 92 for 2 in 15 overs before rain interrupted the encounter. Earlier, England handed over maiden ODI cap to Olly Stone who came in as a cover for Liam Plunkett.
The EU is opposed to any specific cut-off date.
They will go on through the weekend and diplomatic sources in Brussels expect Raab to come on Monday if there is a deal.
Cabinet Brexiteers are angry with May's refusal to put a definite end-point on her plans for a so-called Brexit "backstop".
Brussels would also strip Britain of its £4 billion a year European Union rebate even if there was a second Brexit referendum voting.
The Northern Ireland "backstop" has emerged as the main stumbling block in the Brexit negotiations.
The DUP leader has warned the prime minister not to accept a "dodgy" Brexit deal she will later regret.
Downing Street also announced a further 29 no-deal notices would be published on Friday afternoon, setting out what businesses and consumers should do in the event of there being no agreement by 29 March 2019.