If I were a dual citizen of the United States and Italy and had a job in the Untied States that I could perform remotely, could I move to an EU country other than Italy and work remotely for the U.S. company while living in Europe?
How would taxation and health insurance work? Does freedom of movement to work apply if the job is not actually based in the place you hope to move, but could be performed from there?
Taxation will depend on the country into which you move. Some countries have mutual agreements against double taxations, aimed at avoiding charging residents twice. These usually don't include countries which are considered fiscal paradises/too fiscally advantageous, mainly to fight tax evasion. Since you mention Italy I'll use this country as an example. The Italian Finance Ministry website has a page showing a list of all the mutual agreements signed by Italy. The point is that the information you are looking for is likely to be find in the mutual agreements between your country(ies) of citizenship(s) and your future country of residence. Note however that there is a distinction to be made between residence and fiscal residence: the place where you sleep at night is not necessarily the place in which you pay taxes.
However, being a US citizen also means that your revenue is subject to taxation in the US regardless of where you reside. As the IRS puts it:
If you are a U.S. citizen or resident alien, the rules for filing income, estate, and gift tax returns and paying estimated tax are generally the same whether you are in the United States or abroad. Your worldwide income is subject to U.S. income tax, regardless of where you reside.
There are some fiscal advantages you can claim, such as the Foreign Earned Income Exclusion if you qualify for one according to the requirements. My personal piece of advice is that you get an accountant that know about dual citizens, US citizens and taxation in the EU.
As an EU citizen you have the right to reside in another EU country, provided that you are either working or you have comprehensive health insurance and sufficient resources not to be a burden on the host country's social security system. In principle, I don't believe you have to be working in your host country to qualify under the "worker" condition. However, national health insurance / social security schemes are not well joined up, so you may find it easiest to take out private health insurance anyway.
In my personal experience, when I (UK national) was living in Spain and employed by a UK company I didn't have national health coverage in either country: I didn't qualify to register with the UK's NHS because I wasn't resident in the UK, but I couldn't get a social security number in Spain without either a contract from a Spanish company or proof of being self-employed.
Taxes can be interesting in this situation. I was being paid in GBP, but I had to declare my income in EUR so that Spain could tax me. The fun part was the exchange rate: I had to find an official exchange rate for each payday and apply it to the earnings from that payslip. If you're being paid in USD then you'll need to get advice from a local accountant on how you need to calculate income for local tax purposes.
