I am not a medical person, but I remember reading some time ago that some people who do not regularly use olive oil have digestive issues when they travel to a country that uses a great deal of it, and Greece certainly qualifies on that score. With hindsight, I think that might have affected me a bit on my first trip to Europe in 1972, which did not include Greece but did include Spain and Italy. These days, an American is more likely to encounter olive oil in the normal course of things, but I don't think it would hurt to switch to olive oil for general cooking purposes in the period leading up to your trip. Who knows? It might help.
I've been to Greece three times, but never to Hydra. The only water caution I've encountered in guidebooks involved the island of Skiathos, whose tap water is brackish and (at least in the past) considered safe but unpalatable. Everyone was drinking bottled water there during my visit 20+ years ago. I, myself, have never had a disgestive problem in Greece, and I've probably spent 7 or 8 weeks in the country.
There is one other, non-gastro, thing: A doctor on Santorini told my mother that some visitors to that island have respiratory issues due to the volcanic dust. I would speculate that it probably happens only to people who have underlying pulmonary problems. I've never read anything about this is a guide book, but I figure the doctor was in a position to know, because he'd see some of the sick visitors.