The 22-year-old, who has put pen to paper on a four-and-half-year contract at the Britannia Stadium, was granted a work permit after Potters manager Tony Pulis attended a hearing at Wembley.
"It's fantastic news that we have been able to push through the deal," Pulis told the club's official website.
"Brek is a very, very exciting player, and one who we firmly believe could go on to become an outstanding player for this football club.
"He has terrific pace, a very good left foot and is capable of scoring goals. He has all the tools needed to be a success here."
Shea has been capped 15 times by the USA at senior international level.