I swapped out my GTX 670 with a GTX 680 and the new 397.64 drivers. Opened Squad fearing the worst yet again, however, got on the main title screen okay enough to brave the shooting range and finally, with a sighed relief joined a full server with lots of people. It worked for me and I am confused as much as everyone reading this is so this solution might be the reason. I'm going to test this more and if it was a driver issue then I'll swap back to my 670 and try again.
I'm having the same problem as well, before the update my little GFX 670 (Yes that's correct Grandpas GPU) worked seamlessly and now seconds after the loading screen to the main menu it gives me the crash report. Very concerned. Send halp soon.