The former Florida governor told CNN's Jake Tapper on "The Lead" the race is "wide open" -- pushing back on New Jersey Gov. Chris Christie's claim that it's between Rubio and himself.
Bush also took swipes at Florida Sen. Marco Rubio and Texas Sen. Ted Cruz, comparing the two first-term senators to then-Illinois Sen. Barack Obama in 2008.
"There's nothing in their past that suggests they can lead. They might be able to lead, but how do you know?" Bush told Tapper.
He pushed back on characterizations of him as part of the Republican "establishment." He acknowledged his father and brother were presidents and his mother is one of America's most popular women, but his record suggests he's willing to take on the status quo.
"I disrupted, when I had a chance to govern and serve, I disrupted the order in Tallahassee for the betterment of 18 million people," Bush said.
Bush also blasted Trump over his complaints about a Cruz mailer that accused recipients of a "voter violation" and instructed them to caucus Monday night.
"Donald Trump lost, and he better get used to it, because it may happen more often than once," he said.
Bush also offered an anecdote to show his dedication to winning the Granite State.
When he read about a woman who said she had been a big fan of him as Florida's governor, but now that she lives in New Hampshire, said she didn't want to "waste a vote" on a candidate who has no shot at winning the state's primary, he called her.
"I actually called that person up and I asked for her vote and I got it," Bush said.
The Tampa Bay Times on Tuesday morning quoted
Marie Stevens, a payroll manager who lived in Florida from 2000 to 2002, saying Bush was "just amazing" as governor, but that "I don't want to waste a vote if he's not going to win or has no chance."
It was Bush's challenge in a nutshell: The race's focus has increasingly been on front-runners Cruz, Trump and Rubio. And the surprisingly strong Iowa third-place finisher Rubio is working to consolidate the establishment vote at the expense of Bush, Christie and Ohio's John Kasich.