Note that an empty string is equivalent to a 'false' value, while a populated one is 'true' (unless it evaluates to a numeric zero). You can also get rid of your !'s by reversing your AND/OR tests and negating the original terms:

sub chkfmt {
   if ((!$DISCPCNT || !$DISCDAYSDUE || !$NETDAYSDUE || $NETDUEDATE || 
+$TERMSDISCAMT) ||
       (!$DISCDUEDATE || !$TERMSDISCAMT || $DISCPCNT || $DISCDAYSDUE |
+| $NETDUEDATE || $NETDAYSDUE) ||
       (!$DISCPCNT || !$DISCDUEDATE || $DISCDAYSDUE || $NETDUEDATE || 
+$NETDAYSDUE || $TERMSDISCAMT))
   {
      print TSERROUT "...";
   }
}
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Note that with all of these negation reversals in my head, I might have messed up your original logic. Take a look at that and see if you can follow it, making corrections as appropriate. Even if I got everything right in the translation, the code above still may not match what you're wanting if your original code was broken to begin with. Hope this helps.