I've previously answered this question for ancient Greece here. While we don't actually hear about left-handed warriors in Classical Greece, the general *right-*handedness of people became a tactical feature in Greek warfare, simply because it made one flank of any marching column or battle formation more vulnerable than the other.

In theory, it should have been obvious to the Greeks that this vulnerability could be countered by deploying some left-handed men on the right end of the line. We don't hear of this, though, so presumably it didn't happen. We do not know why not. However, the Spartans - the only Greeks to practice formation drill - trained to reduce the vulnerability of their unshielded side; their basic formation evolutions included moves intended to turn the face of the phalanx towards the enemy and turn the right flank away. This kind of quick response was no doubt better than being forced to rely on specialist left-handed files on the right of each unit (especially since units varied in depth, and therefore in the number of southpaws needed).

Generally, we shouldn't overstate how much a Greek heavy infantry force was "formation-dependent". Scholars are increasingly stressing that hoplites could function in a range of different tactical roles and circumstances; they did not fight only as part of a whole, and did not depend entirely on the phalanx. Careful cohesion and closed ranks helped if they could be formed, but they were really only essential if hoplites were facing cavalry. Therefore, flexibility was far more important to a hoplite formation than perfect protection. Throughout the Classical period, the equipment of hoplites gets lighter, as men seem to choose mobility over heavy armour. They would not tie themselves to a special group of left-handed warriors just to be a little bit safer when drawn up for battle - especially since the right wing, on which the left-handers would have to be deployed, usually had a tactical advantage anyway (see the post I linked to above).

So, what would happen to you if you were a southpaw in a Greek army? We don't know. As I said, we get no examples. Possibly, like children in schools in the last century, people who tended towards left-handedness were forced to use their right hand anyway. Alternatively, individual lefties could be hidden within the ranks and no one would really notice the difference, except for the local interruption of the shield-spear-shield-spear pattern. In the chaos of close hoplite combat, would anyone really care?