Untested, but this should meet your requirements:

#!/usr/bin/perl
#  Putting all of the zonefiles lised in named.conf into an array
use strict;
use warnings;

my $srce = "named.conf.brian";
my %zonefilenames;

open(my $FH, "<", $srce) or die "Failed to open file $srce ($!)";
while (<$FH>){
  next unless my ($filename) = m/file\s*"([^"]+)/;
  next if $filename eq "db.cache";
  $zonefilenames{ $filename } ="";
}
close($FH);

print "Zone Files: ",join( ", ", keys %zonefilenames)," \n";
 
#  Putting listing of all of the files in ~/zonefiles directory into a
+n array
#  Assigning variable to directory
my $directory = "~/zonefiles/";

opendir(my $D, "$directory") or die "Can't opendir $directory: $!\n";
while ( readdir($D)){
   next if m/^\./; # Ignore things with leading dots
   chomp;
   my $found = exists $zonefilenames{ $_ } ? "" :  "Not found in zone 
+definition";
   print "$_ : $found\n";
}
closedir($D);
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Update : Enabled "strict" and fixed code typos.