1. Why don't you do the replacements in NameReplace instead of pushing all filenames to a global array?
2. I'd use

   s/\bservername(?:\.aa\.company(?:\.zzzz)?\.com)?\b/NEWNAME.\com/gi;
   [download]

   in favour of your 3 replacements.

sub NameReplace
{
     if( $_ =~ /\.(?:html?|cfm|cfml|cgi|js|pl)$/)
        {
          my $name = $File::Find::name;
          open ( F, $name ) || warn "$!: $name\n";
          $data=join '',<F>;
          close (F);
          if ($data=~ s/\bservername(?:\.aa\.company(?:\.zzzz)?\.com)?
+\b/NEWNAME.\com/gi) {
                  if (open (F, ">$name")) {
                      print F,$data;
                      close(F);
                  }
                  else {
                      warn "..."
                  }
          }
     }
}
[download]