Here, without all the extraneous modules, is simple example code. You're not using the Archive::Zip module properly.

#!/usr/bin/perl
use strict;
use warnings;

use File::Spec::Functions qw'catdir catfile';
use Archive::ZIP qw':ERROR_CODES :CONSTANTS';

# here's an example @chunks array
my @chunks = qw(file1.txt file2.txt file3.txt);

#example output file name: output.zip in HomeDir
my $zip_file = catfile($ENV{HOME},'output.zip'); 

my $zip = Archive::Zip->new();
foreach my $file (@chunks) {
   my $member = $zip->addFile($file);                         #add fil
+e 
   $member->desiredCompressionMethod( COMPRESSION_DEFLATED ); #compres
+s file
}

#write ZIP file to disk
my $status = $zip->writeToFileNamed($zip_file);
die "Unable to write to file '$zip_file' if $status != AZ_OK; #error c
+heck
[download]

Note that I never open my own file handle, the module takes care of all that for me.