I think this is an older topic, but as people are still referencing it I figured I'd post an update. macOS security has only gotten stricter since this post and will get more so upon the launch of Big Sur later this year (2020).
Use the Linux package of any of the programs and move the folder OUT of the Downloads folder (suggest the users Applications folder would be a good place). New security features will prevent most JAR files from loading under Downloads.
After that, from
a Terminal window, you'll need to launch them as follows:
java -jar MegaMek.jar
java -jar MegaMekLab.jar
java -jar MekHQ.jar
If Java is installed and configured, you should also be able to double click on the JAR files directly to launch.
As of yet, a reliable way to launch them as a packaged app has not been found as Apple keeps increasing security in macOS.