A United Boeing 737-800, registration N37298 performing flight UA-1612 from Indianapolis,IN to Denver,CO (USA), had remained at FL280 below RVSM airspace enroute due to the autopilots being inoperative. The crew decided to divert to Colorado Springs,CO due the weather conditions in Denver advising Colorado Springs they were an emergency divert and needed 1nm visibility for landing (Denver was a quarter of a mile). The aircraft was on short final to Colorado Springs' runway 17L the crew needed to go around at decision height due to being unable to establish visual contact with the runway. The crew attempted another approach to runway 17L but again needed to go around. The crew inquired about weather conditions in Denver and Pueblo,CO (USA), located about 33nm south of Colorado Springs. ATC reported Denver was still pretty much the same, Pueblo not much better. The crew decided to divert to "GUC" Gunnison,CO (USA) with Pueblo as an emergency diversion option, climbed to FL200 and landed safely at Gunnison about 46 minutes after the second go around in Colorado Springs.