Nearly all places in Greece have Western-style toilets, including hotels and restaurants. Public toilets at transport terminals (bus and train) sometimes have Ottoman/Turkish squat-style toilets.
Public toilets tend to be limited to airports and bus and train stations with the very occasional one in tourist-heavy town centres. Cafes are the best option if you get caught short, but in tourist-heavy places, it's polite to buy something for the privilege.
The Greek plumbing system can't handle toilet paper; apparently the pipes are too narrow and anything larger than a postage stamp seems to cause a problem. Toilet paper etc must be placed in the small bin provided next to every toilet.