The ice at Nenana this year is a little thinner than usual, which has led some to speculate that the breakup will be early, although the historical correlation between early April ice thickness and the breakup date is very small. A much more important factor is the temperature in April - and in early May if the breakup is late. Of course, for those playing the Classic, the submission deadline of April 5 makes it impossible to know with confidence how temperatures will evolve in the last 2 or 3 weeks prior to breakup. However, I find it interesting to note that the long-range temperature forecasts from the National Weather Service's CFSv2 model are modestly skillful and therefore allow for a slight edge over random chance in the guessing game.
The chart below shows the breakup date (day of the year, vertical axis) against the forecast temperature for April, according to the CFSv2 forecasts as of April 1 of each year since 1982. The forecasts for prior years are obtained from the historical "reforecasts" provided by NOAA - these are the retrospective forecasts used to calibrate the operational forecasts of today. I extracted the forecast average temperature over the Tanana River drainage basin upstream of Nenana, with the idea that upstream temperatures are probably important as well as temperatures at Nenana itself. There is a modest negative correlation between the forecast temperature and breakup date (R=-0.65), which is not particularly impressive but is better than nothing. (Note that the actual April temperature in Fairbanks predicts the breakup date with R=-0.85, and the forecast predicts the actual with R=+0.77.)
Simple linear regression allows for a crude estimate of the potential effect of the expected temperature on the breakup date. This year's forecast temperature for April is 23.0 °F, or 1.9 °F above normal, which translates into a breakup date of April 30, or 2 days earlier than the 1982-2013 mean. Of course, this is only a slightly better guess than simply taking the mean; the error distribution on the regression indicates that the probability is about 66% that the breakup will be earlier than May 2.
From a subjective standpoint, I would guess that the breakup will be earlier than the forecast indicates, because there is very little snow on the ice (see below). This is unusual for the time of year, and with no reflective snow pack, the sun will be unusually effective in melting the ice. My best guess for breakup is between April 23 and April 28.