Boris Johnson refused to answer questions about police being called to his flat as he faced Tory members at a hustings in Birmingham.
Moderator Iain Dale was booed for the line of questioning regarding the on Friday night but Tory MP Vicky Ford felt he was right to ask.
"Well I find myself with @IainDale again," she tweeted. "He's right to ask questions that are on people's minds."
The Tory leadership favourite was also criticised by opponent Jeremy Hunt supporters for his lack of detail on Brexit.
Johnson said Britain must be prepared to leave without an agreement on October 31 - and said that stance would help the country "get the deal we need".
David Morris, MP for Morecambe and Lunesdale and a supporter of Hunt, tweeted: "Disappointing from @BorisJohsnon in Birmingham no substance in his answers on how he will leave the EU. Worrying lack of detail in answer to any of the members questions #ConservativeLeadership #HasToBeHunt."
Greg Hands, another Hunt supporter, said Johnson's outlook appears to have shifted since the BBC debate.
He tweeted: "Boris's promo video at the Birmingham hustings repeats the line on Brexit to come out 31st October 'deal or no deal'. Which is rather different to what he told MPs in the BBC debate that it was only 'eminently feasible'."
Jacob Rees-Mogg, a supporter of Johnson, tweeted: "An excellent performance from @BorisJohnson at the first hustings. Strong, clear answers and we will leave by 31st October."
Asked how he would go about Brexit negotiations between now and October 31, Hunt said: "A wise prime minister makes choices on the basis of the options that are actually available in front of them at the time.
"Now we don't know what the situation will be on the 31st of October, but if there is no prospect of a better deal, if the European Union have been as inflexible as you say, and if no deal is on the table, then I will leave without a deal because we must keep our promise to the British people.
"But I would do so with a heavy heart, because of the risks to businesses up and down the country and the risks to the union.
"But if we have to, in the end I would do that. But the reason that I haven't made the pledge that Boris has made is because I think it's very important that prime ministers keep their word.
"It is entirely possible that Parliament will rule out no-deal between now and October the 31st."
Hunt added: "I want to get out as soon as possible and I believe that I have the negotiating skills to give us a better chance of getting a deal that can actually get through Parliament."
