The former Foreign Secretary accused the government of being spineless after a bruising clash with the EU last week in Salzburg.
Mrs May failed to get EU bosses to sign up to the Chequers Plan, and returned to Britain accusing them of being unfair and warning them of the chance of “no deal”.
Britain crashing out of the EU on March 29 without a deal is feared to lead to major disruption and shortages of food and medicine.
Johnson laid out his “better plan for Brexit” in a column in the Telegraph newspaper calling for the dismissal of Chequers, a sentiment echoed by Rees-Mogg on Question Time.
The two setting out their stalls against Mrs May comes just days before the Conservative Party Conference in Birmingham – which could be a make or break week for the Prime Minister.
“The Chequers proposals are the worst of both worlds”
Boris quit the cabinet back in July in protest at the Chequers plan, and has been a thorn in the side of the PM ever since – despite battles in his personal life.
And Rees-Mogg has been steadily growing in popularity as he represents the hard Brexiteer wing of the Tories – chairing the powerful Eurosceptic faction The European Research Group.
Johnson accused the Government and civil service of a "pretty invertebrate performance" in negotiations and said there had been "a collapse of will by the British establishment to deliver on the mandate of the people".
In a 4,500-word column, Mr Johnson also set out an alternative vision which he says would make Britain "rich, strong and free".
As a priority, the former London mayor urged the Government to ditch Chequers and negotiate a Canada-style free deal that would "fulfil the instruction of the people”.
BREXIT: Boris Johnson and Jacob Rees-Mogg could bring down Theresa May over Chequers
Mr Johnson said: "Overall, the Chequers proposals represent the intellectual error of believing that we can be half-in, half-out: that it is somehow safer and easier for large parts of our national life to remain governed by the EU even though we are no longer in the EU.
"They are in that sense a democratic disaster. There is nothing safe or 'pragmatic' in being bound by rules over which we have no say, interpreted by a federalist court.
"The Chequers proposals are the worst of both worlds.
"They are a moral and intellectual humiliation for this country. It is almost incredible that after two years this should be the opening bid of the British government."
BORIS JOHNSON: Brexit has the Tories at war over how to break away from the EU
JACOB REES-MOGG: The Tory toff is one the biggest voices against Theresa May's plan
Mr Rees-Mogg echoed Boriss comments, saying negotiations had been "badly conducted” – with the Tory toff also regarded as a potential future Prime Minister.
He said: "I think we have let the European Union make the running in negotiations, we agreed to their establishment of the terms of negotiations and the timetable of the negotiations.
"I think this has not been impressive and I think Chequers is not leaving the EU, it is remaining bound to an EU rule-book which is interpreted by the European Court of Justice and the EU's rejected it anyway.
"So Chequers may not not even be a dying duck, it may be slightly more Count Dracula in that it seems to get up at night and walk abroad, but it doesn't seem to have much life in the sunlight.”
THERESA MAY: The Prime Minister may not be able to survive collapse of our Chequers Plan
Jon Trickett, Labour's shadow Cabinet Office minister, said Mr Johnson's column revealed the Tories had only "internal warfare" left.
He said: "As Boris Johnson says, there has been 'a collective failure of government.
"But not just on Brexit, on housing, education, community safety, on our NHS, public services and utilities.
"It's clearer than ever that the Tories have run out of ideas. All they have left is internal warfare and continued austerity."
BRITAIN DIVIDED: Brexit has become a issue which has split the country down the middle
Lib Dem MP Layla Moran said Mr Johnson's comments were nothing more than a "half baked, sloppy rant".
She said: "The blue on blue in-fighting has already kicked off ahead of Tory conference.
"But while Johnson is being paid hundreds of thousands to belch out the first thing that comes to mind on page, there are businesses up and down the country wondering whether they'll still exist after Brexit.
"It's a disgrace that he can't take the most important decision facing this generation seriously."