I have a java webstart application that is properly signed and working on all OSes. However running it on MAC requires the user to circumvent the Mac security features, as the app is not signed with an Apple Developer ID.
I want to know if and if yes, how I can use my Apple certificates to sign the jar such that it can be run without problems on Windows and Mac.
Unfortunately this is a not possible, and is a duplicate of: How to sign Java applet with Apple Developer ID
You can however use a tool to convert your jar into an app bundle, which can in fact be signed for gatekeeper (not for App Store, but to bypass the launch restriction). See http://www.jemchicomac.com/how-to-convert-a-jar-file-into-an-app-for-osx/