The law that you are referring to is the Foreign Account Tax Compliance Act (FATCA), which was enacted in 2010, and which for the most part becomes effective on July 1 of this year. FATCA requires foreign financial institutions to report to the IRS information about financial accounts held abroad by U.S. taxpayers. It also requires foreign business entities in which U.S. taxpayers hold a substantial ownership interest, and foreign trusts in which U.S. citizens are beneficiaries, to report certain information to the IRS.

Specifically, with reference to your customer’s (probably misplaced) concerns, beginning July 1, FATCA will require foreign financial institutions to provide annual reports to the IRS on the name and address of each U.S. citizen (and green card holders), as well as on the largest account balance in the year and total debits and credits of any account owned by such person. If a foreign financial institution fails to comply with these reporting requirements, the IRS will impose a 30% withholding tax on the gross funds transferred to the foreign financial institution. FATCA also requires U.S. citizens (and green card holders) who have foreign financial assets in excess of $50,000 to file a special form with their tax filing (this requirement has been in effect since 2011). Non-compliance with this reporting requirement could subject a U.S. citizen (and green card holder) to a penalty of up to 40% of the foreign-held assets.

You can read a plain language explanation of the law on Wikipedia here. You can find more information than you probably want to know on the law on the IRS website here and on the Dept. of Treasury website here.