Most Greek troop types were referred to simply by the equipment they carried, or by a generic term like pezoi (infantry) or stratiotai (soldiers). Elite troops might be termed logades or epilektoi (picked ones), and later special units might get a name of their own, like the hieros lochos (Sacred Band). Light infantry sometimes got the less literal label of gymnetes (naked ones), which is probably closer to what you're getting at. But none of these are nicknames.

Warriors who had served for a long time under a particular commander could sometimes be named after him. The men who fought with Brasidas in Thrace in 424-422 BC were still known as the Brasideioi (men of Brasidas) in 418 BC, even though Brasidas himself had been dead for years. The Greek mercenaries who fought with the Persian pretender Cyrus in 401 BC went down in history as the Kyreioi (men of Cyrus) - although they are now mostly remembered as the Ten Thousand.

We get more interesting examples from Aristophanes' comedies. In the Acharnians, the old men from the town of Acharnai near Athens refer to themselves as Marathonomachoi (Marathon-fighters), referring to the glorious Athenian victory at Marathon that even they were probably too young to have taken part in themselves. It is possible that the hoplites of Athens more generally liked to call themselves by this nickname. In the Lysistrata, a similar group of old men exhorts itself to fight like they used to, and these old men refer to themselves as leukopodes (white-feet). The reference is probably to the dust-covered feet of warriors marching in the sun. (Ancient commentators mistook the word for lukopodes (wolf-feet) and offered suggestions as to why Athenian hoplites might have called themselves by such a name.)

Finally, Xenophon once refers to the Theban army as pyr pneontes (fire-breathers), but this is probably meant to mock them. The nickname occurs when the Thebans advanced on Sparta itself, and one of Sparta's kings led a charge of just 100 men against them:

At that moment the fire-breathers, the men who had defeated the Spartans, the men who were altogether superior in numbers and were occupying higher ground as well, did not withstand the attack of the troops under Archidamos, but gave way.

-- Xen. Hellenika 7.5.12