Cain says he would not support Perry as GOP nominee

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"I happen to think that puts the children of illegals ahead of citizens," Cain said.

Cain's unwillingness to support Perry in a general election contrasts with his views of other Republican candidates with whom he differs. The former businessman said, for instance, that he could support former Massachusetts Gov. Mitt Romney, whom Cain he has criticized for implementing a statewide healthcare law that required all Massachusetts residents to have insurance.

"Well, I wouldn't say I couldn't be able to support him because he said his first order of business would be to repeal ObamaCare," Cain said.

But Cain also cautioned that he would withdraw that support if he sensed Romney wavering on that promise.

Other Republicans have shied away from similar statements, arguing that even a Republican with whom they have divergent views would be preferable to President Obama. Speaking at the Republican debate last weekend, Romney — responding to a question about which of his fellow panelists he might tap as vice president — said that any of his fellow candidates would qualify.

"You want someone who without question could become the president of the United States. These people could all fill that — that position. Any one of them would be a better president than what we have now," Romney said.