Use an array instead of multiple $out_reqN variables with the index in their name.

Also, -text doesn't expect a reference, but a normal string.

...
my @out_req = (
    '1. Carpet Plot Data',
    "2. Laminate Properties",
    "3. Laminate Stresses",
    "4. Laminate Strains",
    "5. Circumferential and Radial Stresses/Strains",
    "6. Displacements",
    "7. Strains Per Ply",
    "8. Stresses Per Ply",
    "9. Failure Criteria Per Ply",
    "10. Automatic Search for Failure",
);

for my $label (@out_req) {
    $chk_{$n} = $out_req_frm -> Checkbutton(-text=> $label,
    ...
}
[download]

%chk / %chk_ would also more naturally be arrays (not hashes), as the index $n is just a counter.

See perldata and perldsc.