Looks like AMD was unhappy with some of the GLSL shaders. This is now fixed and Swanky Paint should work fine on AMD cards.
If you're still having problems, do let us know and we'll get back on the treadmill.
Source: Steam Community (opens in a new tab)
Build and release context
Recorded Steam release date:
Within one calendar day of this post (UTC). The listed date does not establish an exact launch time.