Steve Baker, who also quit as a Brexit minister, urged her to step up preparations for a "no deal" scenario.
He said Ireland is not interested in division or separation and there are a number of elements in the UK's White Paper on Brexit that could form the basis of negotiations "that might provide a framework" for how these matters might be dealt with in the future.
It is unclear what sort of European Union market access financial firms in Britain will have after the transition period ends, prompting many banks and insurers to have new hubs up and running in the bloc by next March to avoid potential disruption.
The UK could refuse to pay its £39 billion divorce bill to Brussels if it does not get a trade deal, Brexit Secretary Dominic Raab has signalled.
Mr Hunt's trip to Berlin for talks with counterpart Heiko Maas marks the start of a frenetic round of diplomatic activity for Theresa May's Government as the Prime Minister seeks to build support for her Brexit strategy.
Ms McDonald said she came away with "no sense of reassurance" about British plans for the border issue after meeting Mrs May this morning.
The immediate danger for May appeared to recede as pro-Brexit lawmakers held back from calling for her to go, emphasizing instead the need for a new policy.
Still reeling after her Brexit plan triggered the resignation of senior members of her cabinet, May flew to Northern Ireland on Thursday for a two-day visit to see up close the troubled British region's frontier with EU-member Ireland.
"On the future economic partnership, the white paper (blueprint) raises three sets of questions for which we are expecting answers", Barnier told a press conference after consultations with European Union ministers.
She said: "The economic and constitutional dislocation of a formal "third country" Customs border within our own country is something I will never accept and I believe no British Prime Minister could ever accept".
Thirty-eight percent of people would vote for a new right-wing party that is committed to Brexit, while nearly a quarter would support an explicitly far-right anti-immigrant, anti-Islam party, the poll found.
With time running short and little sign of May quelling the revolts in her party, there has been renewed discussion among Brussels diplomats and officials about whether a deal can be done by October, or at the latest December, to allow parliaments on both sides to ratify a withdrawal treaty before March 29.
But he questioned the UK's plans for a common rulebook as it only referred to goods checked at the border, not areas like pesticide use, adding: "How can we protect consumers in Europe?"
"We need choices and decisions, clarity and legal certainty", the French negotiator said.
Foreign Secretary Boris Johnson, the face of the Brexit campaign, is also staying put to defend his vision of the divorce, according to a person familiar with his thinking.
In her speech, May fired several shots at Brexiteers, warning Eurosceptics that the United Kingdom can not "wash our hands" of the Irish border issue by leaving it to the European Union to sort.