In short, no. As you mention in the docs, gen-class state fields will be public final fields. However, that field can be (for example), an atom that is statefully modified by the implementation methods. In general, we do not give much weight to encapsulation in Clojure - instead preferring to make things visible, but "safe" (via immutability). To quote Rich from https://clojure.org/reference/datatypes, "encapsulation is folly".
gen-class is not a general purpose DSL for generating all possible Java classes. It is a tool to generate classes in a certain style in line with Clojure's aesthetics.
Another path however to something along these lines is to use a deftype, which can have private mutable fields, exposed by implementing interfaces or protocols inline. The deftype fields can have meta of either ^:volatile-mutable or ^:unsynchronized-mutable, both of which will become private fields.