There is no definite article in the Greek of Hebrews 9:27
When the 'the' is put in, one might think of it as 'the great white throne judgement'
But that is not what it says
A man dies once, and there is 'krisis' after that
Reincarnationists believe that as well, upon death one has judgement which determines their future state - whether a new incarnation or some other realm and another incarnation later on.
So Hebrews 9:27 is
Not
A clobber verse against reincarnation, I have heard people use that verse as a quick dismissal of reincarnation, but it is simply not justified in the Koine Greek
I don't know much about Buddha, I don't think Jesus was the same person, but the bible does not deny reincarnation, in fact, think of the disciples asking "who sinned, this man or his parents, that he was born blind?"
Jesus's answer was that it was neither one, but the concept of it possibly being the man himself that sinned, causing him to be born blind, was not negated.
The disciples obviously viewed it as a viable option.
The bible neither confirms nor denies reincarnation, and there is the puzzling passage where Jesus says John the Baptist was Elijah -- yet Elijah as Elijah is recognizable at the transformation, when Moses who did die and Elijah who never died were there.