Most satellites have some sort of a safe mode, a mode the satellite enters when it is having difficulties with certain things. The exact nature of the safe mode varies from satellite to satellite, but in general, it is a lower power state, thermally stable and power positive, turns off science instruments, and relies less on absolute pointing.
I can say that most safe modes can be by-passed in some way or another. For instance, New Horizons will disable it's safe mode for a few weeks around the Pluto flyby, and if it doesn't succeed in doing that, the mission was pointless. However, safe mode did trip during the Jupiter flyby, which was deemed non-critical.
As far as what caused Juno's safe mode during flyby, I can't help but suspect it's related to this statement, as quoted here:
Juno's flyby was the first and only time it would enter a solar eclipse and be forced to rely on its battery for power
Bottom line, I suspect that the satellite determined it didn't have enough power to make it through the eclipse with all of it's instruments on full power, which is one of the most common safe mode triggers for solar-powered satellites. If this is the case, the probe should be up and running again in no time. This is the only eclipse of the mission, most likely the batteries aren't specified to last much longer than this eclipse, and perhaps either the batteries degraded with time, or else they weren't specified correctly. In either case, the spacecraft still is working and overall positive, so it should be fine.
Tildalwave provided me with the system requirements, which includes a nice section about safe mode. Essentially what happens is the data rate drops, the attitude may be compromised, and as a result, lower gain antennas are used for communication.