Every non-binary person might prefer the use of different pronouns. There is a list of neopronouns, some of which are becoming more widely used, but I think if you were to use a specific subset to refer to any non-binary person it could be seen as offensive and presumptuous. Also, a lot of people these days would probably look at you askance if you were to use neopronouns, as they would be completely unaware of their existence. If it’s clear you’re talking about a single person, yet you’re using they/them pronouns it is clear that you’re being polite, thoughtful and inclusive.
Consider the following examples and even test them out with people you know and see the reactions you get:
I saw Charlie yesterday. Ze was sad because ze lost hir mobile phone.
I saw Charlie yesterday. They were sad because they lost their mobile phone.
Edit: The above works only as an example in the context of the question, and in no way does it mean that you should be using common neutral pronouns if a person has specifically asked for a different set of pronouns to be used when referring to them.