if (system(@args1) != 0) {
                logEntry ("No repeater.pl is running, attempting to re
+start");
                $repeater;
        }
[download]

I don't think that's doing what you think it does. As far as I can tell the bare $repeater doesn't do anything. You probably want something like

        if (system(@args1) != 0) {
                logEntry ("No repeater.pl is running, attempting to re
+start");
                $repeater = Proc::Background->new('/etc/squid/repeater
+/lib/repeater.pl');
        }
[download]

Also, Proc::Background has functions for checking whether or not the process is still running, so you may want to use those instead of shelling out to pgrep:

        if (!$repeater->alive()) {
                logEntry ("No repeater.pl is running, attempting to re
+start");
                $repeater = Proc::Background->new('/etc/squid/repeater
+/lib/repeater.pl');
        }
[download]

Of course that would not detect a process that was already started when your watcher begins. (But in that case you'd currently have 2 running processes anyway since you unconditionally start one with the initial Proc::Background->new call.)