Run Jenkins jobs on Linux under a User
Yes, it is necessary to set JENKINS_USER variable, but in another file.
Look here http://blog.manula.org/2013/03/running-jenkins-under-different-user-in.html
Have a look at the Started-By Environment Variable plugin.
https://wiki.jenkins-ci.org/display/JENKINS/Started-By+Environment+Variable+Plugin
EDIT:
If you are just asking how to set up Jenkins to run under a different user in Linux it depends on how you installed it. I never set up Jenkins on Linux yet (but I can use google). Have a look at https://wiki.jenkins-ci.org/display/JENKINS/Installing+Jenkins for different setup options. I had a look at Installing Jenkins as a Unix daemon and at step 7 you need to setup a configuration file. Change the last line in that file (RUN_AS_USER=jenkins) to contain the user of your choice.