I 100% concur here that OP is mistaken, and must be doing something wrong. I was working heavily with the JSON objectified bool last week writing my GPS software. Here's an example that illustrates what the json->perl conversion looks like, and the resulting output before/after re-assignment:

use warnings;
use strict;

use Data::Dumper;
use JSON;

my $json = '{"true": true, "false": false}';
my $perl = decode_json $json;

print Dumper $perl;

my $t = $perl->{true};
my $f = $perl->{false};

print "orig true:  $perl->{true}, var true: $t\n";
print "orig false: $perl->{false}, var false: $f\n";


__END__

$VAR1 = {
          'false' => bless( do{\(my $o = 0)}, 'JSON::PP::Boolean' ),
          'true' => bless( do{\(my $o = 1)}, 'JSON::PP::Boolean' )
        };

orig true:  1, var true: 1
orig false: 0, var false: 0
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