But European Council president Donald Tusk ripped up Mrs May's blueprint, saying it risked the integrity of the EU single market and the border, in a move widely regarded as a humiliation for the prime minister after two days of talks.
"Insulting her on social media, getting to these standoffs where you are calling people liars and so on is not the way we are going to get a solution to this hard situation", Hunt said.
"Until we do, we can not make progress", she said.
"This is the moment to do what is right for Britain", the Sunday Express quoted May as saying.
"We have to be careful, like two hedgehogs who love each other", Juncker said.
Relations between the EU and Britain have always been a bit prickly, but EU Commission chief Jean-Claude Juncker said Friday that the Brexit negotiations are beginning to resemble the courtship of two hedgehogs.
"From the very beginning of the negotiations we have been focused on finding a deal that will minimise the damage resulting from Brexit".
The Prime Minister also angrily rejected calls for a second referendum, after the prime ministers of Malta and the Czech Republic publicly backed the idea.
With some opponents of Brexit actively working with Brussels to undermine the Government's negotiating position, she said Conservatives needed to come together in a spirit of national unity and "do what is right for Britain".
With barely controlled anger, she warned that she would never accept any agreement on the Irish border issue which led to the "break up of my country" and made clear she would walk away rather than accept a "bad deal".
Hours earlier in a strongly-worded statement at Downing Street, Mrs May said her plan had been rejected by Mr Tusk without him properly explaining why it was unacceptable or offering an alternative.
One of the aides was said to have been overheard asking another Tory strategist: "What are you doing in November - because I think we are going to need an election".
He added: 'Brexit shows that it is not easy to leave the EU.
But she said fault lay more on the United Kingdom side: "I would accept there is a level of acrimony and frustration that the European Union has not moved".
"She said Brexit meant Brexit when she didn't know what it meant".
The biggest single obstacle to a deal is the need to maintain an open Irish border.
Europe is insisting on a fall-back plan, a "backstop", that would keep Northern Ireland in the Customs union under EU rules while a future trade relationship is negotiated.
But he said European Union leaders had agreed that the proposals needed to be redrawn: "The suggested framework for economic co-operation will not work, not least because it is undermining the single market".
"We stand ready", she said.