Since I use squirrelmail at this end, I don't see how squirrelmail can interfere with sendmail delivery. Squirrelmail uses http and imap protocols. The only time squirrelmail interacts with sendmail (smtp protocol) is when it submits e-mail from the user, but this is (by default) done through localhost based on the following entries in /etc/squirrelmail/config.php.
$useSendmail = true;
$smtpServerAddress = 'localhost';
$smtpPort = 25;
$sendmail_path = '/usr/sbin/sendmail';
With the above in mind, I would start using tcpdump/ethereal to diagnose the problem you describe. Based on your description, this sounds like a firewall related issue.