Meghan said Mr Juste’s death had left her “heartbroken and reflective”.
Penning a rare latter, she wrote: “We shared, amongst many things, a commitment to animal welfare, and a deep love of rescue dogs.
“In fact, it was Oli and his fiancé Rob who helped care for my rescue dog, Guy, when I had just moved to the UK and he was recovering from a debilitating accident.
“They loved him as though he was their own.”
Meghan Markle
Phil Noble
Her letter, shared with the Independent, came after Prince Harry addressed Invictus Games competitors.
Vice-captain Kelly Leonard said after his address: “I was nearly in tears when he gave us a team talk. It was really rousing.
“The message was about what Invictus is about, that resilience and that journey.
“It was really wonderful to see him.”
It was the Duke of Sussex’s first public appearance since missing Prince Philip’s memorial.
The letter comes after Meghan tried to trademark the word “archetypes” after revealing it will be the name of her first series of podcasts.
The application was made at the US Patent and Trademark Office last month by Archewell Audio, one of several businesses founded by Meghan and Harry.
The word “archetypes” derives from ancient Greek and is believed to have first entered the UK in approximately the 1540s.
Archewell’s application says the word would cover a long list of entertainment services "in the fields of cultural treatment of women and stereotypes facing women".
Meghan Markle and Prince Harry
Daniel Leal-Olivas