The first place I would look for that kind of situation is to make sure the user owns their mailbox folders, and then check the permissions of the mailbox folders on the system.
On my system the mailbox files are owned by the user and in the user's group as well, and I have mailbox permissions that are rw for the user and that's it. My user's mailbox files are located in their home directory so they already have permissions to access the directory.
So to duplicate the way my system is set up would be:
1) Mailboxes in user's home directory, usually in a subfolder called Mail.
2) chown user.user mailbox files (INBOX.Drafts, INBOX.Sent, INBOX.Trash, etc)
3) chmod 600 mailbox files (same as above)
Note that if you change where the mailboxes are located you will also need to go back thru squirrelmail config and change the path to the directory where they are.
Hope that helps.