I decided to remove dovecot and switch to courier-imap after all. The following worked to solve this issue:
chkconfig --del dovecot
chkconfig --levels 235 courier-imap on
Now squirrelmail works fine! Rebooting brings up all the services just fine. If anyone else has this problem, just cut and past the code above and you should be ready to go. At least it will work for Redhat systems others may have to figure out the command to use to add services to load at boot.