I prefer this:

use strict;
use warnings;
use Net::SSH2;

my $ssh2 = Net::SSH2->new();
$ssh2->connect('server');
$ssh2->auth_password('username','passwd');
my $shell = $ssh2->channel();
$shell->blocking(0);
$shell->shell();

print {$shell} "egrep -l $searchStr * 2>&1 | tee ~/myfile.txt\n";
# <$shell> works, but <$hash{$shell_box1}> does not, so
my @results = readline($shell);
[download]

Error-trapping left as an exercise for the reader