JaZZ, I don't know anything about your amp, but the fact that it uses bipolar junction transistors (BJTs) is nothing special in itself indicating that it uses a balanced drive output. BJTs are very versatile semiconductor devices used in all kinds of devices including many op amps. Bipolar refers to the configuration of the three doped regions inside.
OP, as said by others, headphones and speakers do work the same way, and they're wired the same too. (Well, to be more precise, powered speakers have their own amplifiers internally, etc., but the main difference is just the level of the power input and output.)
I think you should just consider the voltages in the system rather than the currents because the voltages are easier to understand. In the end, the current through the headphone drivers is just the voltage across them divided by their impedance anyway.
What the headphone drivers do is convert the electrical voltage across them into vibrations matching that pattern. The voltage across them is the audio signal (which is whatever the source is sending), and it changes rapidly over time. The audio signal is mathematically just the sum of sinusoids at different frequencies with different amplitudes, changing over time. Therefore, half of the signal will be a positive voltage, and half will be a negative voltage. As others have said, the average of the signal, aka the DC component, should be zero or else the headphone drivers will be damaged. Since this signal is changing positive to negative, you could say that it is alternating.
Here is what some music looks like, as taken from a clip I loaded in Audacity. The top is the left channel output, and the bottom is the right channel output. Note that the whole thing may be scaled up or down by multiplication by the amplifier, so this doesn't completely correspond to the voltage seen across your headphone drivers. However, the shape of the voltage will be the same as what you see here:
Again, this signal is the voltage seen across the headphone drivers. In other words, the top line represents the difference between the L audio signal and ground, while the bottom line represents the difference between the R audio signal and ground. Ground is at a 0 signal level, so the difference between the signal and ground is just the signal itself. The ground can be shared and physically connected together because it's just 0. It's the same reference point for both L and R audio signals.
The current from point A to B is the voltage difference between A and B divided by the impedance between the two points. When the signal and thus the voltage is positive, the flow of electrons will be in one direction. When the voltage is negative, the flow of electrons will be in the opposite direction.
Crosstalk is the phenomenon of what's on one channel being reproduced on the other channel. So what one channel sees with crosstalk is the correct signal added to an attenuated copy of the other channel's signal.
Hopefully this explanation is more clear to you.