We have a client that is leaving their current Hosting company and wants to move to our hosting facility. We have recreated their DC,an Exchange Hub, and Exchange Store. We have exported and imported all of their recipient mailboxes and flipped the switch and all mail flow is now going through our hosting facility. The problem lies in that I can not figure out how to move the public mailboxes out of the old exchange server and import it into the new server. Both are Exchange 2013 and running on windows server 2012. Any help would be greatly appreciated. 

You're looking at a third-party product.  MigrationWiz does public folders.  There are other cloud-based migration solutions, but I don't know whether they support public folder migrations.