Can I use MirrorSync to Sync two servers and then use a serial number to generate the invoice number on only one of the servers.

If I only allow users on Server 1 to use the feature for generating an invoice with a serial number and prevent  users that connect to server 2 from using the script that generates new invoices. Will this work?

You should read the Write-back section of the documentation: 

http://docs.360works.com/index.php/MirrorSync_advanced_topics#Write-back_values_for_user-visible_numbers

My plan is to MirrorSync two servers in two different countries so I will not have any users working Off line. They will be either connected to Server 1 or Server 2. If I allow Server 1 users to create the Invoices and block Server 2 users from creating invoices will this solve my problem with the Serial numbers?

The write-back concept applies to all syncs, not just offline users. Using write-backs would allow you to create invoices on either server, and the invoice number on server 2 would be blank until the sync runs, and then invoice number gets assigned on server 1 and then written back to 2.

But the short answer is yes, preventing new invoices on server 2 should also solve the problem.

That is what I was hoping, Thanks