if you can not open JD how do you expect to update ? you need to follow directions.
please try and fix your rotten apples,
terminal or file explorer,
find your way to /Applications/JDownloader/ look for location of jdupdate.jar, think its content/resources/something/ and run jdupdate.jar, I assume you can do that with a double click or right click, open/run with java. from terminal 'java -jar jdupdate.jar'
raztoki
edit
@Xaphier
thanks for you post (just beat me), can you please provide terminal debug logs please.
java -Xmx512m -jar JDownloader.jar -debug
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