most everything will be open on June 2 in Florence
Ciao jdevins and welcome to our DT forum!
June 2nd is indeed a holiday, it is the day of the Republic. I called the Florence museums ticket office this morning and was told that actually all of the museums that are generally open on Tuesdays will remain open. The Uffizi will be open for sure, as will the Accademia (where Michelangelo's David is) and the Pitti Palace. City museums like Palazzo Vecchio should also be open - I am looking on the official website for those and none should be closed on June 2.
Actually, because it is a holiday many people will take Monday off to make it a 4-day weekend --- thus the whole city will be welcoming many Italian tourists as well. Expect the city to be crowded, you might be better off reserving a ticket at the Uffizi beforehand if you want to go. Friday May 1 was a holiday, some people were reported on the local news as being in line for 4 hours at the Uffizi.
Because of this you should have no problem finding most restaurants and shops in the historical center open as well.