I see various spellings of the same, which one is correct?
I have considered that the spelling might differ if it is British or American English, but as English isn't my native speak I have no clue.
For AmE, you should go to the best N. American dictionary, and use the version there:
: not having existence
Merriam-Webster Unabridged Dictionary
CORPUS OF CONTEMPORARY AMERICAN ENGLISH
thus there's a heavy use of both, but I'd follow the M-W U.
BrE: Non-existent used to be British spelling, but a couple of years back they did away with the hyphens of 16,000 hyphenated words.
AmE: the answer above is the valid answer, just one word: nonexistent The American Heritage Dictionary 5th Ed. confirms this.
So it appears the Standard Usage in both side of the Atlantic is one unhyphenated word.