One thing to keep in mind is that you cannot opt out of Italy’s public healthcare system. The public health system is pretty good, but standard vary between the north and south. Many choose not buy health insurance cover. In Italy it is seen as a bit of a luxury.
However, if you can afford it and you want to ensure quick access to the best healthcare, then private health insurance is a good idea.
Personally, I wouldn’t buy health insurance cover. But if you want peace of mind…go for it!