SQLyog is a MySQL client. It can connect to any MySQL server running on any OS. You can migrate your MySQL to Linux platform and still you should be able to work with SQLyog.
I am not sure by what you mean by customization to SQLyog. Do you mean the connection details that you entered in SQLyog? SQLyog is a Windows client. Though it can run under WINE emulator in Linux. More details on running it under WINE can be found at: http://www.webyog.com/faq/31_71_en.html