No, that would muddy things in my opinion, like it did to JavaScript. Instead, have more operations/functions for dealing with them in a more "normal" way, so that we can say "WHERE x <> 5" and get results one expects. I'm not sure the syntax, and my drafts would take a lot of time to explain. To give a taste, maybe have something like "WHERE ~x <> 5" in which the tilde converts x's value to the type's default, such as a blank in the case of strings.
If the different reasons for "emptiness" matter, then usually it suggests the need for a "status" column of some kind so that queries can be done on the reasons. I'd need to study domain specifics to recommend something specific.