I guess for people like that it comes down to something like this:
Barack Obama is black. Also, his middle name is "Hussein" and he went to school Indonesia, which means he's an Arab (since Indonesia is an Arab country). And Arabs were responsible for 9/11. So, Barack Obama can't possibly be a real American. He's something different, something un-American, something alien, therefore maybe even something dangerous.
And whether he literally was born abroad or is literally ineligible (even being born abroad wouldn't make him ineligible, of course) is irrelevant. He is not a real American and doesn't belong here.
Then they're surprised why Asians and Hispanics don't vote Republican more.