The developments have triggered speculation that May could face an imminent leadership contest.
Her spokesman said she had welcomed the new members of her cabinet and they had discussed the publication of a white paper policy document on Britain's future ties with the European Union and stepping up preparations for any no-deal outcome to the negotiations with Brussels. Following her statement, she went to a private meeting of the Conservative Party's 1922 Committee, where she was cheered warmly on arrival and when she left. She also told her party that "to lead is to decide".
After the pro-Brexit politician was the star of the successful Leave campaign, it was expected he would make a bid at becoming prime minister when a humiliated David Cameron stood down.
He was enthusiastic about Johnson, calling him "a friend of mine".
"I think she (Mrs May) showed authority on Friday, has provided a lot more certainty in respect of the British approach towards the Brexit negotiations and I expect her to see that through now", Mr Coveney said.
Meanwhile, Ms Caulfield wrote: "'I cannot support the direction of travel in the Brexit negotiations which, in my view, do not fully embrace the opportunities that Brexit can provide".
Matt Hancock replaced Hunt as health secretary, while Attorney General Jeremy Wright was appointed as minister of digital culture, media and sport, Hancock's old job.
"Top-level cabinet resignations usually spell trouble for a government and Boris Johnson's might well have led to a leadership challenge, but May appears to have emerged unscathed from a meeting of her party, for now", said Al Jazeera's Jonah Hill, reporting from London.
"Before asking other employees to consider relocating, we will wait for further political and regulatory clarity in line with our overall post Brexit strategy", according to the memo. He in turn is succeeded by Attorney General Jeremy Wright, who is replaced by Geoffrey Cox.
Mixed emotions in Scotland as England prepare for semi-final
Martinez said the loss to France really hurt them, disclosing that the team's dressing room after the match was a sad atmosphere. Didier Deschamps' squad definitely has a thousand faces, and they could all be smiling at the end of this World Cup .
Serena Williams through to fourth round at Wimbledon
The South African eighth seed reached the third round when he completed a four-set victory over Italy's Andreas Seppi on Thursday. The 38-year-old American - last year's runner-up at Wimbledon - was the oldest entrant in the women's singles field.
Former Malaysian leader arrested amid multibillion-dollar graft probe
Justice Department says a total of $4.5 billion was looted from 1MDB, some of allegedly landed in Najib's personal bank accounts. Malaysia's Prime Minister Mahathir Mohamad says that authorities have an "almost ideal case" against Najib and his allies.
Fianna Fail's Brexit spokesperson Lisa Chambers said the double resignation was of "grave concern" but urged the British Government to continue with its plan to publish the white paper this week.
The final two then go to a postal ballot of all party members, with the position of leader, and in this instance prime minister, going to the victor.
"We'll make everything possible if the president wants to do something", he said.
Mrs May moved quickly to appoint the staunchly anti-EU Dominic Raab as the new Brexit secretary - a shrewd move that should calm some Brexiteer nerves. Davis slammed the agreement, which promotes a "soft exit" from the European Union versus a swift, clean break, as giving away "too much away, too easily" to E.U. negotiators. In addition to the prime minister's office, the three most critical cabinet positions - Treasury, the Foreign Office and the Home Office - are safely in the hands of those who voted to remain in the EU.
Nigel farage has threatened to return as the leader of the UKIP unless Brexit is "put back on track" and he also accused May of betraying the people.
"The Government now has a song to sing".
Labour deputy leader Tom Watson said: "Theresa May's Government is in meltdown".
"What the prime minister is proposing is a way in which we can ensure we don't have. friction with our trade with the European Union. and here is a practical way in which we can do that", Gauke said.
"May is fighting for her survival", Mujtaba Rahman, a managing director at Eurasia Group, said in a research note.
Addressing Parliament on Monday, May told MPs that she and the two ministers were at odds over "the best way of delivering our shared commitment to honoring the result of the referendum".