Just a follow up for others who might face this issue in the future.
I got the old host to zip up the client's mail folders and send them to me. So I tried uploading them via FTP to the client's MAIL directory on my server. Unfortunately, this did not work. When the customer logged in to SquirrelMail, they did see the folders (typically located in the left column), but they were only accessible by clicking OPTIONS, but the enclosed emails were not viewable. Hmmmm.
So I brainstormed a bit and came up with a plan. I asked my client to manually create the folders in his account. Once that was done, I then uploaded all the mail to those manually created folders. Voila. It worked!!
So the trick is to manually create the folders in the SquirrelMail account, and then you can upload all the archived emails.