Johnson is the overwhelming favorite to replace Theresa May as Conservative party leader but has limited his public appearances, including skipping a candidates' TV debate.
The former foreign minister and one-time London mayor has been accused of not having a detailed plan to take Britain out of the European Union.
He is also under growing pressure following a scandal over an argument with his girlfriend that led to a police visit to their home last week.
"I am not interested in debating Boris's private life," Hunt said in an article in The Times, arguing he instead wants to quiz Johnson on his Brexit plans and other ideas.
"A new prime minister needs the legitimacy of having made his arguments publicly and having them subjected to scrutiny.
"Only then can you walk through the front door of No 10 with your head held high instead of slinking through the back door, which is what Boris appears to want."
Urging Johnson to attend a TV debate proposed for Tuesday evening, June 25, Hunt added: "Don't be a coward Boris, man up and show the nation you can cope with the intense scrutiny the most difficult job in the country will involve."
Sky News said Johnson had so far declined its invitation to take part in Tuesday's head-to-head debate, and that it would reissue an invitation for a debate on July 1 instead.
Screams, shouts, bangs
Johnson, 55, was involved in a loud altercation early Friday, June 21, at the home of his 31-year-old girlfriend Carrie Symonds.
Neighbors called police after hearing screams, shouts and bangs at the south London property, shortly after Johnson had secured his place in the final run-off to become prime minister.
The couple, who were spoken to by officers but face no further action, have reportedly fled the home following the incident and a handful of anti-Johnson protesters gathering there Sunday, June 23.
Photographs emerged Monday of the couple smiling and holding hands at an undisclosed countryside location said to have been taken on Sunday.
When asked about the incident at a Conservative grassroots event the previous day, Johnson said: "I don't think people want to hear about that kind of thing."
He used his regular column in the Daily Telegraph newspaper on Monday to reiterate his commitment to delivering Brexit at the end of October, insisting he would not "bottle" it.
But an increasingly vocal band of Conservative MPs have indicated they could topple a Johnson government to prevent the country leaving the bloc without an agreement.
In comments for a BBC documentary to be aired Monday, defense minister Tobias Ellwood said around a dozen Tory lawmakers, including other ministers, would support a vote of no confidence to stop a no-deal Brexit. – Rappler.com