https://www.irs.gov/individuals/international-taxpayers/fore...
"If you are a U.S. citizen or a resident alien of the United States and you live abroad, you are taxed on your worldwide income. However, you may qualify to exclude from income up to an amount of your foreign earnings that is adjusted annually for inflation ($92,900 for 2011, $95,100 for 2012, $97,600 for 2013, $99,200 for 2014 and $100,800 for 2015). In addition, you can exclude or deduct certain foreign housing amounts."
IANAL, but aren't there treaties between the US and all EU countries that avoid any double taxation? You have to pay income tax wherever you live most of the time and conduct your business. You never have to pay income tax twice, in two countries. Unless you obtain your income mainly in the US, you should be exempt from paying US taxes. This is independent of citizenship.
Sorry if I'm missing something, I thought (and have been told) that this is how it works, so I'd be happy if someone else with more knowledge could clarify this.
However I suspect that in the case of a US citizen, that doesn't mean you don't have to officially declare your tax status/income to the IRS each year. Hence the parents worry that he's going to have to trawl back over his entire working life to 'prove' that he doesn't owe any back taxes.
IANAL, but am from the UK, lived in the US for a few years and have had to deal with moving back and forth.