Do OpenGL polygons need to be convex?

Yep.

Well..... I dont know about when you are deep into GL, (or DirectX), but Ive found that it works like that. What I do is have a "Shape" made from convex bits of shapes.
Its just max2d commands though. (And its only solid colours)
http://www.blitzbasic.com/Community/posts.php?topic=66251#740265