I spent some time yesterday cleaning up some oddities on my own system and doing a fresh install of EVE Online.
This managed to clear up the initial blue screen I was getting.
Now I am getting a different error when I run two clients. A different blue screen.
Now my computer blue screens with an error related to the ha10kx2k.sys.
But at least I am not alone in that regard. There is a long thread in the EVE forums about other people having this issue. While there is a lot of speculation about what might be causing it, a little research seems to indicate that it is related to the drivers for my sound card, a Sound Blaster Audigy 2.
I suppose that is some progress.
Since my motherboard has sound built-in, I will try yanking the Audigy 2, uninstalling the drivers, and see how that flies.