There are ways to avoid repeat searches of each line but this should get you started.

#!/usr/bin/perl
use strict;

my $count = 0;
my $namefile = $ARGV[0] || 'names.txt'; # default if no argument
# names.txt
#ServerName1
#ServerName2

# get list of computers to search for
my @computers;
open FILE,'<',$namefile or die "Could not open $namefile : $!";
while (<FILE>){
  chomp;
  s/^\s+|\s+$//g; # trim spaces
  next unless /\S/; # skip blank lines
  push @computers,$_;
}
close FILE; 

my $file = "computernames.txt";
# computernames.txt
#<Answer type="string">ServerName1.FD.net.org</Answer>
#<Answer type="string">ServerName2.FD.net.org</Answer>
#<Answer type="string">ServerName3.FD.net.org</Answer>

# search text file
open IN, '<',$file or die "Could not open $file : $!";
while (my $line = <IN>){
  # repeat line search for each computer
  foreach my $search (@computers) {
    if ( my ($name) = $line =~ /($search[\.\w]*)/ ){
      print "Name ; $name\n";
      ++$count;
    } 
  }
}
close IN;

# result
if ($count){
  print "$count matches\n";
} else {
  print "No matches found\n";
}
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