After two days of trying to get through with Disney, I got the information needed from representatives (lots of time on hold to confirm information).
1. To board the ship in Rome, Disney follows CDC guidelines so fully vaccinated is two full doses. Booster not required. Italy has relaxed rules entering the country, and those not boosted (but vaccinated with two doses) require a negative test.
2. For our sailing, there are 5 Greek ports. On May 5, Greece fully opened up and do not require vaccinations or tests to enter the country. From the Greek government site too:
*IMPORTANT NOTE* The requirement for completion of Passenger Locator Form (PLF) by visitors from ALL countries is no more mandatory from March 15, 2022.
Furthermore, as of Sunday 01.05.2022, all travellers arriving in Greece, regardless of their country of origin, are no longer required to display a valid certificate of vaccination or recovery from COVID-19, or evidence of a negative test result from SARS-CoV-2 infection (PCR or Rapid Antigen)
3. Naples Italy: Right now that would require a test to disembark. Agent stated that most likely there will be testing on board as some will not be boosted. The cruise line is deciding this a week out for each sailing. For example, he said the current transatlantic crossing has testing on board for people who want to visit ports of call. This is our only port of call that would require testing. Rules may change by June 27.
This will be moot for my kids if boosted 2 weeks prior to cruising. However good information to have.