FET-based H-Bridge for bi-color LED
Here is how I drive bi-color LEDs:







simulate this circuit &ndash; Schematic created using CircuitLab

Use a logic family that can source or sink 20ma or whatever you need to drive the LEDs.  I actually don't use inverters (that solution was closest to your circuit), but a serial to parallel shift register (like a 74xx591) to drive 4 bi-color LEDs using the 8 outputs.  For two output pins you end up with 00 -> off, 01 -> red, 10 -> green, 11 = off.