I would optimize the processing in order to avoid wasting time and memory.

• You only need to hold two input lines in memory, since there are no dependencies to other lines.
• One random shuffle should be enough to make it random.
• I would check whether the input file contains an even number of lines, that is when retrieving a pair of lines, check first, if the DNA sequence line is present.
• $final_seq should be initialized to the empty string in order to avoid a warning on an 'undefined' value.

use strict;
use warnings;
use List::Util 'shuffle'; # Idea from http://www.perlmonks.org/?node_i
+d=199901

my $input = shift @ARGV;
open(IN, '<', $input) or die "Can't read multifasta input genomic DNA 
+file $input : $!\n";

my $destination = $input."_1MBWindow_ListUtilshuffle.fasta";
open(OUT, '>', $destination) or die "Can't write to file $destination:
+ $!\n";

my $window = 1000000; # hard coded for shuffle window to be 1MB i.e 10
+^6

my ($seq_id, $seq);

# process every alternate line with ID (and its corresponding sequence
+ in next line)
while (defined($seq_id = <IN>) {
    if (!defined($seq = <IN>) {
        last;
    }
    chomp $seg_id;
    chomp $seg;
    my $final_seq = '';

    for (my $i = 1; $i <= length $seq; $i += $window ) {
        my $s = substr ($seq, $i - 1, $window);
        my @temp_seq_array = split //, $s;

        @temp_seq_array = shuffle @temp_seq_array; # using the List::U
+til module AND Shuffles EACH window!!!
        my $rand_shuffled_seq = join ('', @temp_seq_array,);
    $final_seq .= $rand_shuffled_seq; # concatenates the shuffled DNA 
+seq to the 3' end of the previous 1MB fragment
    }
    print OUT $seq_id, "\n",$final_seq,"\n";
}
close IN;
close OUT;
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