Controlling 4 steppers + 2 servos with an Arduino MEGA

Is it better to use directly motor shields or the H bridges,

Most motor shields can only support two steppers so you are probably going to have to use external stepper drivers.

Does the stepper ever fails?

Each stepper motor has a limit to how much torque they can produce. You must make sure that the torque your motor produces is more than enough to overcome any friction in the cube. Generally the larger the stepper motor, the higher the voltage, and the higher the drive current, the higher the torque. Current will be limited by your driver and by how much power the motor can dissipate.

Quad stepper driver, $65, 1 Amp:

Single stepper driver, $13, 2 Amp: