I am wondering if they would experience less symptoms of PTSD due to growing up training to be warriors, and having war more deeply engrained in the culture than in American culture.
While I'm no expert on this subject, I did a course on Classical Greece at university taught by Jason Crowley who has written a lot on this subject and Classical Greek warfare generally; fortunately I remember enough to write a decent answer.
Jason's argument is that no, Classical Greek hoplites did suffer PTSD because they they were subject to far fewer psychological pressures than modern soldiers. Let's go through the reasons in turn for both the Greeks and the modern soldier:
Culture and religion: The Classical Greeks were raised in a society when warfare was considered normal and indeed even necessary to express one's masculinity. The best way of doing this was by fighting as a hoplite and getting into the thick of the action, rather than as a sailor, skirmisher, or a cavalry soldier. In addition, the Greek pantheon had numerous gods focusing on war or martial prowess. Contrast this with modern soldiers who are raised in societies that generally shun war and where war is a lot less prevalent. Practically all modern religions condemn violence and ascribe punishment in the afterlife for those who engage in it.
Organising of armies: Classical Greek armies were essentially city-state citizen militias. While they were only part-time soldiers, they did go out to fight every campaigning season. Because of political organisation being being based on city-states, army sizes were relatively small. Furthermore, armies were organised into units based on city districts, meaning that soldiers would be fighting alongside neighbours and friends, thus increasing unit cohesion. Armies also rarely campaigned far from home. Compare this to the modern national army, where soldiers are recruited from across countries with potentially astoundingly large populations. Furthermore, soldiers are regularly reassigned in and out of units, damaging unit cohesion. Modern armies also often fight incredibly far from their home country in unfamiliar, foreign environments; this links to...
Technology: The level of technology in Classical Greece results in most fighting consisting of close combat/melee. The benefit of this is that soldiers fight face-to-face, and know exactly who it is that they're fighting (usually other Greek citizen militias). Technology also restricts travel to being by foot or by ship, leading to most Greek armies fighting in Greece and therefore familiar territory. Modern technology however means that soldiers are often fighting an enemy they can't see; long-range guns, artillery, aircraft, etc. Additionally, because of technology affecting logistics, soldiers can be in constant combat readiness for months at a time without a proper rest. As already mentioned, transport technology results in soldiers being able to campaign around the world, far from home and many things that would be familiar.
Further reading:
Jason Crowley, 'Beyond the Universal Soldier: Combat Trauma in Classical Antiquity', in D. Konstan and P. Meineck, Combat Trauma and the Greeks (London, 2014) pp. 105-130.
Jason Crowley, The Psychology of the Athenian Hoplite: The Culture of Combat in Classical Athens.