So, when the mission starts, and the guy tells you that a dead gun is a good gun the game crashes to desktop. Any ideas?

Corrupted installation maybe, make a backup your savegame, uninstall and try re-downloading, preferably with the GOGdownloader, and re-installing the game.
I played through the game without issues last year so it should be possible.

I have the same problem, re-installation has not fixed it and I can't get any further without the crash reoccurring. Please help!

Same problem.
Seems like I need to skip that mission with cheats.
Should crash at these boring EMP missions and I wouldn't comply with skipping the missions.

First, you must right-click on the Aquanox desktop icon and go to properties option. Inside the target field press the space bar once and then type -redrum -stendek.

[F10]- Auto mission complete
[F11]- Manual mission complete

EDIT
or not.
I just clicked right mouse button on Aqua.exe, checked properties and set to Windows XP compatibility (SP3), then turn off desktop fancy stuff.
When mission start and King start talking - click left mous button to boost the cutscene.
Voila, it worked.
I have Win7 x64.