“The Prime Minister would do well to tell Mr Gibb to get back in his box - or go back to the BBC,” Boles added, referring to Gibb’s former employer.
On Monday, Boles quit the party after his efforts to find a solution to the Brexit deadlock were rejected by MPs.
Boles said he was resigning the Tory whip “with immediate effect” because the party “has shown itself to be incapable of compromise”.
He said on Twitter that he would continue to sit in the House of Commons as an “Independent Progressive Conservative”.
The next day, he said May’s Cabinet is “the worst in recorded history” and no serving minister should be PM after Brexit.
The Grantham and Stamford MP said ministers had been “selfish”, were guilty of following blind “dogma”, had been “cowardly” and had “failed to lead”.