could you please post the exact SQL Developer version you use? If it's not 3.2.2 (3.2.20.09.87), please update to that release. There have been a few bug fixes that also concern APEX Listener configuration since the initial 3.2 release.
If you are already using the latest release, please post the exact settings you want to use for your connection, and also the JEE container you use to deploy APEX Listener, if you are not running it in Standalone Mode.
Stumbled across your question when I was troubleshooting something similar: 503 Service Unavailable for REST calls with ApEx Listener 2.0.2 DB 12c. My issues started in the RESTful services but I would also get a 503 error when trying to connect to the Listener via SQLDeveloper.
My problem was with the apex_al.xml and apex_rt.xml files generated by the java -jar apex.war setup process. Once I fixed them by hand the 503 errors in SQLDeveloper went away too.
Not sure if this is related to your problem?
Take a look at this post where I come clean on how a typo in the first pass of the java -jar apex.war setup process was the root of my problem: 503 Service Unavailable for REST calls with ApEx Listener 2.0.2 DB 12c
You'll see that I had the wrong connect string details inside of the xml files. Specifically the apex_listener/apex/defaults.xml file had the wrong value for <entry key="db.servicename">.
My first fix was to override it in the apex_listener/apex/conf/apex_al.xml and apex_listener/apex/conf/apex_rt.xml files. What I should have done is scrap my whole apex_listener directory and start over with the java -jar apex.war setup process, as that would have effectively fixed the typo I had in my apex_listener/apex/defaults.xml file.
Hope this is helpful,