TML and Wazza both have it right.
As far as I can tell, these are the only Goldwater-Hillary counties outside the South (I also threw in Florida).
It's amazing how these counties have shifted over the past half-century. Coconino County, Arizona, used to be a solidly Republican county before becoming a Democratic bastion in the 1990s, probably because of the increasing minority population and the liberal shift in Flagstaff. Whitman County, Washington, is now the only Democratic-leaning county east of the Cascades, and its shift can be explained by its large college population. Alpine and Mono Counties, are of course both ski resort counties, and John Kerry flipped them back in 2004. Ski resorts in general are now in the Democratic column. It's especially astonishing considering that Alpine County was once a
extremely Republican county, giving over 90% of the vote to Warren G. Harding in 1920, for example.
Orange and San Diego Counties, of course, can be explained by demographic changes, the alienation of wealthy and educated suburbanites from the Republican Party of the present era, and by the changes brought about due to the end of the Cold War. Obama flipped San Diego in 2008, and Clinton of course, was the first Democrat to win Orange since Franklin Roosevelt in 1936. Gallatin County, Montana, can probably be explained by the college and liberal influx that has occurred there. Teton County, Wyoming, is another resort county, and is now the only Safe Democratic county in that state. John Kerry flipped it in 2004, like Alpine and Mono (though unlike those two counties, Teton did vote for Bill Clinton because of Perot).
DuPage, Kane, and DeKalb Counties, the "collar counties" of Chicago, were in 1964 much more conservative and whiter then they are now, and from the 1850s until Obama won them in 2008, were Republican strongholds. Gadsden County, the only majority-black county in Florida, can be explained by civil rights; 1964 was the year before the Voting Rights Act was passed, and blacks were still disenfranchised in the Florida Panhandle. This also explains Leon County. And as for the remainder of the Goldwater-Clinton counties in Florida, they can be explained by demographic shifts. Orange and Osceola were heavily white back then, and the Hispanic influx which ultimately flipped them to the Democrats had not yet occurred. Palm Beach and Broward Counties were much less diverse as well, had not yet experienced the influx of Jewish and Northern liberal voters, and were dominated by conservative retirees and transplants of the kind that rule SW FL and the Space Coast now.