Burnham, an avid follower of the sport, said he was "heartbroken" to learn of Boston's death, just four days after the funeral of the former Wigan great's wife, Lady Joan.
"A giant of a man in every way. Thank you, Billy, and rest in peace," said Burnham.
Born in Cardiff to a father from Sierra Leone and a mother from Irish stock, the Welsh ace Boston, one of 11 siblings, grew up playing rugby union before switching codes to sign for Wigan in 1953 at the age of 19 where he was to prove a wing wizard.
He went on to score an incredible 478 tries in 488 appearances for the club during the 1950s and 1960s.
Australian audiences often witnessed the best of Boston, who scored 24 tries in 31 appearances for Great Britain and was part of the side that won the 1960 World Cup, scoring a try in the final as they beat the Kangaroos 10-3 in Bradford.
After playing just a few league games, Boston was selected for the 1954 GB side that went to Australia and New Zealand, becoming the first Black British player to tour down under, and he proved a sensation with his 36 tries.
In all, Boston scored 571 tries in his career - a total surpassed only by Australian world record breaker Brian Bevan in top-class rugby league.
But Boston was more than just a great sportsman. He was a pioneering figure in many ways, a Black athlete who broke down barriers by becoming a universally popular figure in the more diverse post-war world in Britain.
Boston, who was to become a legend in Wigan, received a knighthood from King Charles at Buckingham Palace last year, becoming the first rugby league player to receive the honour in the sport's 130-year history.
Burnham wrote on X: "Heartbroken to hear my friend Sir Billy Boston has died.
"Billy was one of the greatest rugby league players of all time but also one of the kindest and most down-to-earth people you could ever wish to meet."
Wigan coach Matt Peet said: "Billy Boston is woven into the fabric of this club.
"Those fortunate enough to have seen him play tell me he was simply the greatest. His achievements, the man he was and the standards he set will continue to inspire everyone who pulls on this shirt."
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