Opposition Labour leader Jeremy Corbyn said the Cabinet ministers had "jumped the sinking ship".
In a remark addressed in particular to Mr Johnson, Mr Hague said: "Being a romantic on this issue is all very well but is of no practical use to the country".
"The government now has a song to sing", he said.
"I believe this will help the government stick to the promises it made".
"It's a step towards a much softer Brexit". "This policy will be bad for our country and bad for the party".
The Conservative leader is at risk of suffering more damaging resignations with the publication of her White Paper today, which sets out the new Brexit policy in detail.
The UK is set to leave the European Union on 29 March 2019, after the 2016 referendum in which people voted by 51.9% to 48.1% for Brexit.
"It takes 48 Conservative members out of a total of 60 to trigger a leadership challenge; however, we are unlikely to see the challenge happen because there is not yet the belief that they have the numbers to effectively oust Theresa May", Hoez added.
Pro-Europeans want to retain close economic ties with the bloc and its market of 500 million people, while some Brexit supporters want a clean break to make it possible to strike new trade deals around the world.
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Mr David Davis replied: "Regret, really".
Former foreign secretary Boris Johnson, who said in his resignation that Britain is headed for the "status of colony", has always favoured a "hard Brexit", whereby the United Kingdom would cut all ties with the EU.
With less than nine months left until Britain is due to leave the bloc, May is sticking to her plan for a "business friendly" Brexit.
In his letter of resignation, he described her readiness to accept a "common rulebook" with the European Union in trade on goods and her proposal of "impractical and undeliverable customs arrangements" as being like "sending our vanguard into battle with the white flags fluttering above them". Her plan splits the difference between a stripped-down free-trade deal and the so-called Norway option of full single-market membership. Working out how to keep the now invisible border free of tariffs and customs checks has been a major stumbling block in negotiations. The timetable increasingly looks overly optimistic, and European Union frustration with British division and chaos is growing.
Mr Coveney said that hard negotiations lay ahead but that the Brexit process has moved from the "politics of parliaments" to the negotiating rooms of Brussels.
"We deliver that Brexit and we do it in a way that protects jobs and livelihoods and meets our commitment to Northern Ireland".
Soon after his appointment, Hunt said he would be standing "four square" behind the prime minister "so that we can get through an agreement with the European Union based on what was agreed by the Cabinet last week at Chequers".
But many eurosceptic MPs are outraged at May's plan, and Davis's resignation letter was scathing. Junior Minister Steve Baker, who worked under Davis, also resigned.
Mr Raab, May's new Brexit minister, will suggest that such criticism was not justified. "It's not even an accidental betrayal, it was planned and plotted well in advance", said Conservative lawmaker Andrew Bridgen.
While May is certainly in a precarious situation, the absence of banner-barer leading the charge for a clear and widely supported alternative position to the Chequers deal makes her relatively safe for now.