The US State Department's website is beyond useless and in many cases, including here, totally inaccurate.

However it doesn't matter for the OP. Your daughter WILL need a new passport.

The rule for the Schengen Area, of which Italy is part, is that third country nationals (including the US) must have passports valid for at least three months beyond the intended date of departure from Schengen (which is very different to six months beyond date of entry as the Dept of State says). With an intended date of departure from Schengen of 12th June, your passports must be valid until at least 12th September.

With tomorrow being a national holiday and so no chance of getting a same day renewal at a passport office, I'm afraid you are going to have to postpone your trip. Call United, explain your case very contritely and politely. The normal case is that you'll have to pay change fee plus difference in fare to reschedule, and that's what you should expect. It's possible that UA staff will take pity and do what they can to help, possibly a waiver of the change fee. There's no guarantee of that at all, and UA owes you nothing, but be polite, explain it's your fault and there's a tiny smidgen of hope. Good luck