Yes, you should have a diode anti-parallel to each transistor to allow the inductive flyback current of the motor to flow, or else it will produce a large voltage spike and burn out your transistors.
THat H-bridge is suitable. Most transistors should work fine since the current is only around 250mA. I would use logic-level MOSFETs (NMOS and PMOS) instead of the BJTs (NPN and PNP) in the schematic. BUt either will work. Just with the MOSFETs you probably don't need the base resistors anymore and the voltage drop is lower across the transistor.
WHy wouldn't the servo circuit work? Seems to me like it should.
Do not PWM this H-bridge or switch it at high frequency- it will have a momentary short-circuit when both transistors are partially on every time you change directions or switch transistors on and you don't want this to happen too often or the circuit will be damaged. This is because the left and right pairs of transistors are being controlled by the same signal. If you wired up the PIC to control each transistor individually (4-pins) you would not have this problem since you could make sure all the right transistors are turned off before turning transistors on to prevent this "shoot-through current".