Short answer: neither. The word you want is nonexistent.
Longer answer: You can actually add a "non" prefix to any word to make up something new, even if it's not in the dictionary. (If you do so, common style says to use a hyphen.)
After having eaten an endless supply of apples, she was pleased to finally be handed a "non-apple".
You might do this because you think that non-apple has a better metre here than something that wasn't an apple.
Typically, when you do this, a common style to indicate you're using it (while acknowledging it's unusual), is to put it in quotes. These are also called "scare quotes"—or, in the real world, "air quotes."
Depending on your dictionary source, "non-existing" is not a word, and, in general, nonexistent would be used instead of it. (Although, if you primarily follow Oxford this is not a concern.)
However, for the sake of argument, let's say that it doesn't exist in any dictionary. Or that the hyphenated version doesn't exist anywhere. There could still be a circumstance in which you want to use the hyphenated version.
"He has no quality of life; he's just existing".
"If you ask me, I'd call it 'non-existing' . . ."
In that exchange, the use of nonexistant is inappropriate because it doesn't express the same sentiment. Nor do nonexisting or not existing have the same meaning. So, for stylistic reasons, the made up word is used.
But I wouldn't call it correct to use a nonexistent word outside of a context such as this, or without signalling (as by using quotes) your acknowledgement of its otherwise incorrectness.
Edit: When I originally wrote this, I had made the mistake of only looking at existing comments and checking Merriam-Webster. I have since updated the answer, given the fact that it the word does exist in Oxford and some other sources.
