Burrell, 61, has become a frequent commentator on Diana and the royal family since leaving the palace after Diana's death in 1997, and was in New York promoting a documentary.
Harry, 35, said in a recently aired ITV interview that he would "not be bullied into playing a game that killed my mum". He and Meghan are also taking legal action against media groups to protect their privacy.
The Duke and Duchess of Cambridge have enjoyed a smoother ride from the Palace and the press because their future is largely predetermined, Burrell said.
The two brothers are often called "the heir and the spare," which Harry hates, Burrell said, and Harry also admitted to ITV that some distance had grown between them.
"Harry said that (he and William) are on different paths. I know what he means by that. William and Kate have a map in front of them. They're headed for monarchy," Burrell said, adding that the Queen and Charles have left Harry to find his way.
"There's no guidance. There's no support. There's no rulebook," Burrell said.
"I hope we don't lose Harry and Meghan (to California) because they are a tremendous asset to the royal family, and to Britain," Burrell said.