I've bought the game recently and I keep getting this black screen and blue square in the top left corner issue after missions. At first it only happened during some missions, but now it happens after everything, including skirmish with AI.
@Jack_Forest - welcome!
Could you repeat this, and after closing your game upload your warnings.log (in [Documents] / my games / Dawn of War III) to https://pastebin.com/ and link it here? Also provide your system specifications.
There you ago https://pastebin.com/pyjG7wsH
It took a while because I was busy and a couple of missions worked fine, for some reason. Now it happened again.
Windows 10 x64
12 GB Ram
GeForce GTX 880M
Run this test for dow 3 and let me know result: https://www.systemrequirementslab.com/cyri