Here is a far faster solution:

#! /usr/bin/perl -w
use strict;

my $length = shift || 6;
unless ($length =~ /^\d*[02468]\z/) {
  die "Length must be an even number";
}
$length /= 2;
my $low = join '', 1, map 0, 2..$length;
my $high = join '', map 9, 1..$length;

for my $first ($low..$high) {
  my $last = int(sqrt($first.$low)) - $first;
  $last = $low if $last < $low;
  my $square = ($last + $first)*($last + $first);
  while ($square < $first . $last and $last <= $high) {
    $last++;
    $square = ($last + $first)*($last + $first);
  }
  if ($last <= $high and $square == $first . $last) {
    print "$first$last\n";
  }
}
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On my system the main execution time seems to be the time to load Perl. It can find both answers of length 10 in under a second.