No.The EU pet entry regulations require you to have a standardized health certificate endorsed by USDA APHIS. This will be enforced at check-in for your flight to the EU. Said certificate is valid for ten days from the date of issue by the USDA veterinarian (so get it as late before your flight is scheduled as possible lest it expires) and acts as substitute for an EU passport that allows your dog to travel to and within the EU during that time.
Assuming you travel to the EU, stay there for a limited time (up to the duration of the time the rabies shot is valid) and then return your dog to the US, there is neither a requirement nor a necessity to visit a vet and have a pet passport issued in any EU country. You'd only need to get an EU pet passport for your dog if you relocated there or were planning a longer trip and wanted to travel around, crossing borders. In this case, you can do this anytime and anywhere you live. I`d also rather get an EU passport for your dog if you planned frequent trips to Europe where you take him/her along, as that saves you from a trip to the USDA every time you fly over. With an EU passport, you will only have to keep the rabies vaccination current by having a vet in the EU(!) administer a rabies shot every three years and document it in the pet passport and you can fly back and forth to your hearts' content without having to worry about USDA trips and 10-day timeframes ever again.

You will be able to transit through Switzerland with no issues even if you choose to leave the airport in Zurich and take doggie for a walk outside, thus entering Switzerland. If you don't leave the airport but stay in the transit zone, you and your dog technically don't enter the country and therefore regulations do not apply.
The health certificate/documentation you get in the US for travel to Italy is valid for all of the EU. Even the form is standardized, the only difference being different languages. The form for Italy (link) is bilingual English/Italian, and as Italian is an official language in Switzerland, Swiss authorities will accept it. Although Switzerland is not a member of the EU, they have fully adopted the EU regulations on pet travel.