Not necessarily the solution, but you should be using

use strict;

# and this really helps DURING development.
use CGI::Carp qw(fatalsToBrowser);
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I tried setting '$ENV{'QUERY_STRING'} = "test";' before the blockquote, and it printed as expected, so not sure yet why(my own ignorance, per usual, but the only way to learn) why $ENV{'QUERY_STRING'} is not being set...

update: I took your original code, and swapped

$exit_link = "$ENV{'QUERY_STRING'}";

# for 

$exit_link = "Perlmonks rocks.";
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after a refresh it pulls up properly. So I gather that you cannot view the cookie contents the same time that you put it, and $ENV{'QUERY_STRING'} is empty(which makes sense now because there was no REQUEST).

update(2); Spelling...