Appearing on CBS' Face The Nation, Thompson said he would work to educate Iowa primary voters about inconsistencies in the records of former Massachusetts Governor Mitt Romney and former Arkansas Governor Michael Huckabee.
Of Huckabee, Thompson said, "Liberal is the only word that comes to mind, when he was governor.
"Cato Institute rated him one of the worst governors in terms of taxes and tax increases," Thompson said. "He wanted to lift the embargo on Castro. Now he wants to close Guantanamo because he thinks it will curry some favor with other nations. It's basically a pretty social liberal policy that he has followed for several years, and hasn't made any bones about it up until recently.
"Now I understand that he's got a tough immigration ad going on, but it's not consistent with his record."
On the question of immigration, Thompson felt being tough on illegal immigrants was the only way to be fair to those who had not entered the country outside normal channels.
"You don't say that once you make it in and once you violate the law, you throw up your hands and say, there's nothing we can do about it so we've got to give you full citizenship, or on the path to citizenship, and set you ahead of all the people who have waited for years and years to play by the rules and to become a part of America."
He said that, over time, stricter enforcement of border security, employment requirements, and clamping down on "sanctuary cities," "there will be attrition over a period of time.
"If they know they can't go back and forth across the border, if they know that they're not going to be working here indefinitely, if they know they're not going to be protected from the law essentially in these cities, that they'll go back on their own accord - obviously, not all of them, and not overnight. But it will move in the right direction."
He also labeled Romney's record as inconsistent.
"Mitt has taken different positions at different times before different audiences at different times in his life and career," Thompson said. "I'm not sure how you would ever determine how Mitt would govern in the future if you looked at his past, in comparison to what he's saying today.
"I've been a strong, consistent, common-sense conservative ever since I first set foot on the national scene. With me, you know what you get. And I am who I am. And it's who I've always been."
He admitted that his performance in Iowa is make-or-break for his campaign, but said that was the case with all candidates, even those with higher poll numbers.
"We need to do well in Iowa, there's no question about that, but others do, too," he told Bob Schieffer.
While Thompson has gotten off to a slow start since announcing his candidacy, he said Iowa is "tailor-made" for his campaign.
"When I'm out on the trail in Iowa, it kind of reminds me of Tennessee. And I'd never run for office before and started way behind and wound up way ahead - you know, just kind of retail politics. And that's what Iowa is conducive to."
Thompson said he would be challenging certain assumptions about Washington's power: the power of the federal government to tax people unreasonably, to spend money it doesn't have, or the power of courts to set social policy.
"It seems like the name of the game is to stick with your game plan, do what you set out to do with your message. Don't try to change your message. Don't try to be something that you're not. And on Election Day in Iowa, the pundits are turning out to be wrong about as often as they turn out to be right."
"Well, that's not the case only in Iowa," Schieffer said.
"I was trying to be kind there," Thompson said.
John Edwards failed to land a key endorsement in the Iowa caucuses, when the Des Moines Register editorial board backed Hillary Clinton - this almost four years after the paper endorsed Edwards in the last Democratic primary.
But the former senator from North Carolina said he had a major disagreement with the paper over dealing with corporate power.
"I think their view was, you work with them, you engage with them," Edwards said to Schieffer. "I think some of the huge corporate power and interests that exist in Washington, you have to take them on and be willing to fight them, if you actually want to bring about change in this country."
In fact, he dismissed the editorial's description of his "harsh, anti-corporate rhetoric" which the paper said "would make it difficult to work with the business community to forge change."
"We just had a fundamental disagreement about that," he said.
On working with the business community to forge change, he said, "If that were effective, we'd already have change.
"I just think the balance is completely out of whack. And I think if we don't have a Teddy Roosevelt kind of president or a Harry Truman kind of president who is ready to take those entrenched, well-financed interests on, it's going to be impossible to bring about change."
The Register non-endorsement comes the same week Edwards is featured on the cover of Newsweek, labeled "The Sleeper." The article describes how, due to arcane caucus rules, Edwards's campaign momentum in Iowa's smaller counties may carry him to victory, despite the stronger showing of Clinton and Barack Obama in the state's larger cities.
"I think what [Iowa voters] are looking for, Bob, is they just want to be treated fairly. They don't want to feel like their government and the way the government functions is being so distorted against their interests. They want a president of the United States and a candidate who will take these interests on to overcome these obstacles."
Edwards did back Obama against barbed criticisms leveled at the Illinois Senator by former President Bill Clinton, who suggested voters would not take a chance on a candidate with little experience on the national stage.
"I think that kind of criticism, just based on a paper resume record, is not legitimate," Edward said. "I think what you need to know is whether he's got the toughness, whether he's got the resilience, whether he's actually prepared, as you picture him in your head or any other candidate, me included - whether they have the command to be commander in chief and the leader of the free world.
"I think that's the test that has to be applied to all of us."
He did note philosophical differences between himself and both Obama and Clinton, in how, in his view, they would work within the current Washington system. "[Clinton] basically says the system in Washington works fine. She defends it. 'We can sort of maneuver our way through this system through the entrenched interests and the lobbyists and so forth and accomplish what needs to be accomplished.'
"I think that if those things worked, we'd have universal health care. We would have a different energy policy. We'd have a different tax policy. We'd have a different trade policy."