Ciao Angela and welcome to our community. While we don't yet know what Covid restrictions will be in place in June, the green pass isn't something that you have to apply for; your CDC white card or an official recovery certificate issued by government or state health authorities is what you need to show when asked for the super green pass in Italy. If your vaccination or recovery certificate has a QR code, it can’t be read by the Italian app that scans vaccination certificates, so your certificate needs to be read manually.
Because you all have boosters, under current regulations your vaccination certificates have unlimited validity in Italy, so you don't need to worry about them expiring. The only reason this might change is if Italy decides to put a 4th dose in place, but thus far there isn't any discussion of this.
Please keep us posted on your travel planning and let us know if we can help you with your itinerary or any other questions. We have rail travel and tour experts here in the community who can help you plan, as well as many seasoned travelers willing to help you with tips and ideas!