(NB, I'm a user, not an admin).
Here's how I would do this:
* Install the JRE system wide (I use ubuntu not Mint, but I'm sure Mint's package manager will have a java version).
* Download just the executable jar file from the download page.
* Unzip it into (e.g.) /usr/local/bin/yed-3.15
Create a file /usr/local/bin/yed, containing:
java -jar yed.jar $@ &> /dev/null
* Do: "chmod 755 /usr/local/bin/yed"
Then as long as your users have /usr/local/bin in their $PATH then simply running "yed" from the command line should work.