But the PM declined to say if he would resign if a postponement – which he has repeatedly ruled out – had to happen.
Mr Johnson has said he would be prepared to leave the EU without a deal, but Labour says stopping a no-deal Brexit is its priority.
The prime minister’s younger brother, Jo Johnson, announced earlier that he was standing down as a minister and MP.
Speaking in West Yorkshire, Boris Johnson said Jo Johnson, who backed Remain in the 2016 referendum, was a “fantastic guy” but they had had “differences” over the EU.
Announcing his resignation earlier in the day, the MP for Orpington, south-east London, said he had been “torn between family loyalty and the national interest”.
During his speech at a police training centre in Wakefield, the prime minister reiterated his call for an election, which he wants to take place on 15 October.
He argued it was “the only way to get this thing [Brexit] moving”.
“We either go forward with our plan to get a deal, take the country out on 31 October which we can or else somebody else should be allowed to see if they can keep us in beyond 31 October,” Mr Johnson said.
He told the audience he hated “banging on about Brexit” but accused MPs of having “torpedoed” the UK’s negotiating position with the EU by voting for a Labour-backed bill designed to block a no-deal exit on 31 October.
The legislation would force the prime minister to delay Brexit until January 2020, unless MPs approve either a new deal or a no-deal exit by 19 October.
However, Labour leader Jeremy Corbyn has accused the PM of having “no plan to get a new deal”.
Liberal Democrat leader Jo Swinson said the prime minister’s comments were “deeply troubling”, and the PM would soon be legally forced to seek a Brexit delay.
The House of Commons rejected Mr Johnson’s plan for a snap election in a vote on Wednesday.
But the government has announced that MPs will get another chance to back this plan next Monday.
The fresh vote on an early election is scheduled just before Parliament is due to be prorogued – or suspended – from next week until 14 October.
Opposition parties are holding talks about how to respond to the prime minister’s call for a mid-October election, amid concern over whether it should be delayed until after an extension has been agreed to prevent a no-deal Brexit on 31 October.
Meanwhile, Brexit Party leader Nigel Farage has warned Mr Johnson that he “cannot win an election, whenever it comes, if the Brexit Party stands against him”.
However, if they were to make a pact during a general election “with a clear policy, we’d be unstoppable”, he told the BBC.
Yvette Cooper, Labour MP and chairwoman of the Home Affairs Committee, criticised the PM for using police officers as a backdrop to his speech.
“This is an abuse of power by Boris Johnson, making so many police stop their training and work to be part of his political stunt,” she said.
West Yorkshire Police chief constable John Robins said he was pleased the force was “chosen as the focal point of the national recruitment campaign” and welcomed Mr Johnson’s pledge to increase police funding.
One of the student officers standing behind the prime minister appeared to become unwell during his speech and question-and-answer session.
Twenty minutes in, she sat down with her head bowed, at which point Mr Johnson apologised and said: “That is the signal for me to actively wind up.”