Hay played 23 internationals for Scotland between 1975 and 1982.
He also took part in three Tests for the Lions on the 1980 tour to South Africa, and he appeared four times for the Barbarians.
The Scotland Rugby World Cup players and management paid tribute to Hay.
Captain Jason White said: "It's obviously very, very sad to hear of Bruce's death.
"I first met him as a member of the Scotland Under-19 squad that he managed at the IRB World Championship in Argentina 10 years ago.
"He was a really genuine guy with the interests of players always uppermost in his mind. On behalf of all the players I extend our sympathies to his family and friends."
Scotland coach Frank Hadden added: "It's terribly sad. Bruce's contribution to Scottish Rugby has been simply immense.
"My personal dealings with him started when he was manager with Scotland Under-19, and he brought a tremendous enthusiasm, commitment and ability - just as he had showed as a player - to that role.
"If there was a more popular man in the game, then I have yet to meet him."
All but one of his internationals and his three Tests for the Lions were played alongside his close friend and rival Andy Irvine, who was Scottish Rugby Union President from 2005 until earlier this year.
Their club careers also ran parallel, with Hay making more than 40 appearances for his Edinburgh district, well over half of them in Irvine's company.
Irvine said: "If he'd been playing today he would have won 50 or 60 caps. For seven years he was involved in every game, playing or on the bench.
"He came from an unfashionable club. He went on to play senior rugby with Boroughmuir, but he always held on to his Liberton roots."
Hay is survived by his wife Lynda and daughter Lynsey.