LabVIEW is natively ASCII, and Unicode isn't entirely supported. Usually customers ask about Unicode because they want to do something with multiple languages. I'm not an expert in Unicode, but from what I understand it is an international standard for encoding text. So if the question is regarding multiple languages, LabVIEW is capable of supporting multiple languages.
As for writing Unicode to a file, you might want to put in a suggestion to the Product Suggestion Center.