The bill will come up for its first debate in the House of Commons on Monday, and mutinous MPs are demanding its withdrawal along with pressing for Justice Secretary Robert Buckland to resign.
Mr Blair and Sir John have accused the government of "embarrassing" the UK.
DUP East Antrim MP Sammy Wilson said that was "nonsense" though.
Their intervention is, however, unlikely to sway Mr Johnson, who's insisted the Internal Market Bill is a necessary safety net to protect the union and peace process - and has defied EU demands to withdraw the contentious clauses before the end of the month.
Labour have indicated they will not support it unless big changes are made.
"They need to explain how Northern Ireland companies having less paperwork undermines the Belfast Agreement". Ensuring the continuation of the existing soft border between the Republic of Ireland and Northern Ireland in the changed circumstances arising from Britain's exit from the bloc is thus paramount for overall stability. "Such a tool is vital to help us fend of predatory behaviour from our nearest competitor".
Brandon Lewis, the Northern Ireland secretary, told MPs last week the proposals, if passed by parliament, would breach worldwide law in a "limited and specific" way - an extraordinary admission that provoked outrage from Brussels.
However, some senior Tories have argued the legislation breaches global law - something the government has conceded - and hence threatens Britain's reputation in the world.
Foreign Affairs Minister Simon Coveney said there may be "limited checks" on goods coming from Great Britain into the region because there is an agreement to prevent the need for physical infrastructure on the Irish border.
Mr Johnson's manoeuvre is "shocking" and "imperils" the Good Friday accord that led to more than two decades of peace in Northern Irelalnd, the two wrote in a joint opinion piece in the Sunday Times.
He later added: "There is no blockade proposed. That is the kind of inflammatory language coming from No 10 which is spin and not the truth".
At present the United Kingdom is part of the European single market, with jointly agreed regulations and standards across the continent.
One major element of the Brexit withdrawal agreement is the section related to ensuring an open border on the island of Ireland to protect the peace process in Northern Ireland.
Mr Johnson has claimed the Bill is required to counter an European Union threat to "blockade" Northern Ireland and "cut it off" from the rest of the UK.
Charles Michel, who heads the EU Council of governments, said Britain's "international credibility" was at stake as both sides battle to unwind almost 50 years of economic integration.
"In fact", the statement said, Brussels "is of the view that it does the opposite".
"We should be getting on with defeating this virus, not banging on about Europe", Sir Keir said.
"I believe it is potentially a harmful act for this country, it would damage our reputation and I think it will make it harder to strike trade deals going forward", Sir Bob told Channel 4 News.
The EU is anxious that post-Brexit Britain could undermine its own food standards, as well as rules on state aid for companies, and infiltrate its single market via Northern Ireland.