If you want to capture a part of the string, you must enclose it in parentheses:

my $separator = qr/(.*)<<<(.*)>>>(.*)/;

# ...

if (my ($before, $name, $after) = /$separator/) {
    print $out $before if defined $before;
    close $out;
    open $out, '>', 'fixed_square' . $name . '.txt' or die "Unable to 
+write to fixed_square$name.txt: $!\n";
    print $out $after if defined $after;
}
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