How to Force 24bit Depth map for OpenGL Linux ?

I believe that OpenGL/Linux build of my game is using low precision depth and color buffer. Windows build looks fine but Linux have a lot of z-fighting, all depth-based image effects are generating artifacts (screen is black when Ambient Occlusion, Ambient Obscurance or Edge Detect is turned on). Also, colors are weird - I think that color buffer also has lower depth precision than DirectX equivalent. Under lowest possible resolution (1280x720 in my case) everything looks fine.

So - is there any way to force OpenGL to generate highest precision depth and color buffers?