On one of the Great Divide journal links, there was an individual who rode a portion of the Divide Ride on what looks to be a cyclocross bike. I sent an email to the group asking if the cyclocross style worked well on the Divide. The response back was that after the first two weeks of the ride, the individual using the cyclocross bike left the trip. The remaining individuals thought that later in the trip through CO and NM, this style of bike would not have worked.
In summary, it looks like it all depends on which part of the Divide Ride you plan on riding. I have been on the parts around Steamboat in CO. On this section, a cyclocross bike would have worked, but the washboard roads would have been very uncomfortable.
Hope this helps.