In the 2016 referendum, they voted 52-48% in favour of Brexit.
During the Brexit briefing, held weekly at Downing Street, senior ministers discussed proposals to ensure the United Kingdom is not indefinitely tied to the European Union via the Northern Ireland "backstop" arrangements.
Cabinet minister James Brokenshire rejected suggestions that final agreement has been reached, saying negotiations are "still very firmly continuing", with 95% of issues resolved.
A spokesman for the department said: "In line with DEFRA, we have issued practical advice for people who wish to travel to European Union countries with their pets in the event the United Kingdom leaves the EU in a no deal situation".
Kuenssberg said Mrs May and cabinet ministers were in agreement about the "horror" of missing the November deadline - a "deeply undesirable" outcome that would tilt the scales towards a "no-deal" Brexit.
But Mr Johnson dismissed this format as "a Christmas present of the finest old Brussels fudge" which would mean the United Kingdom getting "absolutely nothing in return" for its £39 billion divorce payment.
But he made clear that he would not accept an arrangement which gave the United Kingdom unilateral powers to ditch the customs union without the agreement of Brussels.
"From social policy to the environment, from agriculture to industrial standards to immigration, when Brussels says "Jump", the British answer will be 'How high?'" he said.
Brussels would always use the prospect of a hard border in Ireland to keep the United Kingdom "forever" in its orbit, he claimed.
"We are choosing wholesale subjection".
DUP MP Sir Jeffrey Donaldson said Dublin's stance was making a no-deal Brexit likely. Under this plan we surrender control to Brussels.
She said: 'I urge all pet owners who wish to travel immediately after March 29 2019 to consult with their vet as soon as they can.
Downing Street later said there is no further cabinet meeting scheduled this week in order for ministers to reach an agreement on Brexit, although one could be held "at the appropriate moment".
Speaking to the News Letter on Monday, DUP Brexit spokesperson Sammy Wilson said his party would not support "any mechanism that gives a veto to the European Union or the Irish", adding: "We could be stuck with this arrangement forever, this eternal backstop".
Pet owners living in Northern Ireland could be forced to pay hundreds of pounds in veterinary fees and wait up to four months for paperwork before they can travel across the border to the Republic with their dog. "What is missing is a solution for the issue of Ireland". The agreement would "not be done at any cost", her official spokesman said.
There would be an "exit clause" from the customs union in a bid to convince Brexiteers that it is not a permanent arrangement as May looks to secure votes to get the deal through Parliament, added the paper.