I finally got annoyed enough to figure this out. I even tried M$ Entourage and Mozilla Thunderbird to attempt to create first level folders in my IMAP directory. Those solutions don’t suit me since they don’t integrate with Apple’s Address Book.

ANSWER

In the open source IMAP server used by Squirrelmail, Apple Mail gets confused and doesn’t auto map your IMAP folders to Sent, Drafts, Trash, etc.

The tree in the IMAP server was:
INBOX
INBOX/Sent
INBOX/Drafts
INBOX/Trash
etc…

The IMAP folder tree that Apple Mail wants is:
INBOX
Drafts
Sent
Trash
etc…

The ANSWER is to map the INBOX/Sent, INBOX/Drafts, INBOX/Trash by using the menu:
Mailbox -> Use This Mailbox For … -> Sent
Mailbox -> Use This Mailbox For … -> Drafts
Mailbox -> Use This Mailbox For … -> Trash

Here’s a screenshot of the menu option

Then go into your Mail -> Preferences -> Mailbox Behaviors and check to change the server settings to store onto the server.