Warning: software I write is for my own usage, but these are my thoughts...
JavaDB stores data in a folder that is referenced as a url by your program. It is quite distinct from the .jar file that contains your application, so you can simply replace that .jar with your updated version and (provided sufficient testing!) everything is sweet. How you go about replacing the old .jar with the new one is a separate question.
* put it on a web site and let users know it is available. Call me old fashioned, but I like this approach. Your users may be in corporate or other networked environments where deployment of new software just isn't something they are able or permitted to do. Or your users may (like me) want to maintain control of what software runs on their computer, and be assured of and in control of the integrity of their data.
* use
Java Webstart to deploy updates. (There's a link at the bottom for more developer oriented information, including Oracle's Tutorial example.)
* use
installation software (Wikipedia) to automate the business of finding and replacing the old .jar with the new one.
Where things start to get interesting is if you deliberately want to change the content or format of the database itself. (Eg you want to add a new column in some table which you reference in the updated software. Or if the javadb version has changed.) In the latter case - where javadb has changed - I think there is an "upgrade" flag you can use as part of the data url. If the data format has changed I think you have to do this manually: eg on first use, create a new table with the additional column, copy the data from the old table to the new one supplying some default for the new column, delete the old table and rename the new one. [Edit: or use sql to alter the table. On reflection this makes more sense.]