print "Specify full directory of logfile.\n"; 
$file = <STDIN>; 
chomp $file; 
 
print "\nIdentified file is $file\n"; 
print "Is this correct? (y/n)\n"; 
$confirm = <STDIN>; 
chomp $confirm; 
 
 
if ($confirm eq "y") { 
    print "\nCommencing work on $file.\n"; 
    workings(); 
} else { 
    print "Terminating program...\n"; 
     
} 
 
## Subroutine to open & read logfile 
sub workings {  
    print"In Subroutine workings now.\n"; 
    open (LOGFILE, $file); 
    print "Opening logfile...\n"; 
    open (OUT, ">>", 'D:\Temp\test.txt') or die "$!"; 
    print "Opening output file...\n"; 
    print "Checking for string GATEWAY...\n"; 
    while (<LOGFILE>) { 
        if (/gateway/) {  
            $count = $count + 1; 
            print OUT; 
            print "Extracting line ...\n"; 
        } elsif (/TIMESTAMP/) { 
            $count = $count + 1; 
            print OUT; 
            print "Extracting line ...\n"; 
        } 
    } 
    print "\nTotal of $count lines extracted.\n"; 
    close OUT;  
    close LOGFILE; 
} 
 
##Array to contain usernames 
@names = qw(mddinzam khairulz sawaikha caoyx jchew khamshae hartinib t
+eev xiaolh yettynm yussofyu leegm narayam karuppa doraira edanor raza
+liha sambanr jwong lamks linwb rashid ongsp ooisc mohdrosn);
[download]

That prints out the lines containing either or both of "gateway" and "TIMESTAMP". Your if is equivalent to

        if (/gateway/ || /TIMESTAMP/) { 
            $count = $count + 1; 
            print OUT; 
            print "Extracting line ...\n"; 
        }
[download]

However, I was under the impression you want lines containing both. If you want the lines containing both, your if would look like

        if (/gateway/ && /TIMESTAMP/) { 
            $count = $count + 1; 
            print OUT; 
            print "Extracting line ...\n"; 
        }
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As for excluding the listed names, one way is to start by building a regex using one of the following two methods:

my ($exclude_re) = map qr/$_/,
                   join '|',
                   map quotemeta,
                   @names;
[download]

or

use Regexp::List qw( );
my $exclude_re = Regexp::List->new()->list2re(@names);
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The just make sure the line doesn't match that regexp:

        if (/gateway/ && /TIMESTAMP/ && !/$exclude_re/) { 
            $count = $count + 1; 
            print OUT; 
            print "Extracting line ...\n"; 
        }
[download]

perl -ne "print if /gateway/ && /TIMESTAMP/" infile > outfile
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if /(gateway|TIMESTAMP)/

A paren around the collection of alternatives, a | between each alternative.