$ cat Test.java

public class Test {

    public static final Test ONE = new Test() {
        @Override
        public int value() {
            return 1;
        }
    };

    private Test() {
    }

    public int value() {
        return 0;
    }
}

$ cat config.xml

<?xml version="1.0"?>
<!DOCTYPE module PUBLIC
        "-//Checkstyle//DTD Checkstyle Configuration 1.3//EN"
        "https://checkstyle.org/dtds/configuration_1_3.dtd">
<module name="Checker">
    <module name="TreeWalker">
        <module name="FinalClass"/>
    </module>
</module>

$ RUN_LOCALE="-Duser.language=en -Duser.country=US"
$ java $RUN_LOCALE -jar checkstyle-8.41-all.jar -c config.xml Test.java
Starting audit...
[ERROR] /Users/archie/bugs/checkstyle-A/Test.java:1:1: Class Test should be declared as final. [FinalClass]
Audit done.
Checkstyle ends with 1 errors.

The FinalClass module is doing it's job correctly in that it's detecting a non-final class with only private constructor(s), but it's not smart enough to realize that in some situations - e.g., this particular situation where you are defining nested classes within the outer class - it's not possible to make the class final because otherwise those nested classes would be impossible to write, because the compile would fail because you can't extend a final class.

Java allows nested classes to extend a class using a private superclass constructor. So the upshot is that FinalClass should make an exception when the class being checked contains any nested subclasses.