... the script still does nothing.

[root@wsprod01 myperl]# ./test.pl --domain google.com --client google
Usage: ./test.pl --domain companydomain.com --client company [--job-co
+de 12345] [--help].
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In the code shown in the OP, in the statement

GetOptions (
     'domain=s' => \$domain,
     'client=s' => \$client,
     'job-code:i' => \$job_code,
     'help' => &usage()
 ) or &usage();
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the  'help' key is given the value of whatever is returned by a call to the  usage() function. Problem is, this function returns nothing; rather, it calls die to print a usage message, so program execution will never get past the  GetOptions() function call. Is this what you intend?

Updates:

1. If you want to assign the  'help' key a reference to the  usage() function,
       'help' => \&usage,
   would be the correct expression.
2. Added italic emphasis in "... a call to the  usage() function."