While others are talking about morality and ethics, I'll stick to what they can practically do assuming they don't have a moral compass and are just concerned with the actual rules.
First, how did they test positive - was it because they tested in an Italian pharmacy/European lab? If yes, then their passport information will be in the system and authorities would know if they try to leave the country etc specially by plane (by train/car no one checks borders). So most likely they'll have to stay put. Typically there aren't many checks around isolation requirements, but if they try to fly out etc their passport could be flagged.
But, if they used the USA BinaxNow teledoc method for testing then their positive result will be confidential and opaque to the European authorities, so EU won't know about it. This way of testing positive opens up a few options that won't be necessarily possible if they tested positive through a lab with their passport info attached. In this scenario, they could be fully unethical and break the law and get a fake test result to fly back to US (this would also be illegal, so not recommended at all even though some folks do it).
An alternative allowed by the rules way is to fly to Mexico (without a transit/stop in USA), since Mexico does not require a negative test result specially if vaccinated. And then walk into USA through San Diego or El Paso (USA does not require test result at land border for US citizens). This way is fully allowed, but people always say its unethical/immoral so your friends can decide.
A third option is to simply extend your vacation and go through Europe explore other countries (the vaccination serves as a health pass so their positive test result is irrelevant as far as rules are concerned because they have a health pass in terms of their cdc card, they'll be able to travel around Europe).
A fourth option is to keep doing the antigen tests a few more times in case it was a false positive.
Of course the ethical and moral option is to find a 2 week airbnb and isolate, but know that less scrupulous options exist.