One primary resource we have for this time period is the book(s) of the Maccabees. While, as with any primary resource, it's extremely important to take broad statements with a grain of salt, there are certain things we can probably say.

The first thing to say is that overall the Jews felt an attraction to Hellenic culture. They wanted to take part in it, and some things that were preventing them (like circumcisions preventing them from entering Greek baths) were a mark of shame. The book of Maccabees records that Jews fashioned fake foreskins to participate in aspects of Hellenic life (1 Maccabees 1:14-15). I'll note that this action was criticized by the authors of the books. There are also other aspects of Greek culture that play a prominent role in 2nd temple era Judaism. One of the most prominent is the Sanhedrin, a rabbinical council which is derived from the Greek synedrion. This became the de facto governing council for Jewish laws.

Maccabees also reports the other side, the friction between the Greek rulers and the Hebrews. The friction culminated in the war of rebellion which makes up a good chunk of the book and is the basis for the Jewish holiday of Hannukah.

I'll also note with the Persian period, while the Jews did appear to enjoy cultural tolerance, Liverani notes that this is more because the Jews lived in a unimportant region which wasn't really worth Persian oversight. They were much more interested in the coastal trade routes than the inland highlands. So the Jews enjoyed freedom, in Liverani's view, not so much because the Persians were benevolent, but because they were so unimportant that Persia didn't really care.