To follow up on my problems and in the hopes it helps others with a Gigabyte BRIX system with this problem, here's how i solved my problem.
Lukily the Gigabyte BRIX has a HDMI and a mini Displayport connection. After i installed a mini DP to HDMI converter i was able to start up my system and install the Intel driver for the HD 4000 graphicscard. After that everything worked like a charm.