Will my program & the ESHA Database work with SQL Server 2014?

Yes, the ESHA software programs and Database will work with SQL Server 2014.

There is a related knowledgebase article which should walk one through attaching the ESHA Database to an existing instance of SQL Server: Attaching the database

In addition, you may also wish to refer to the article related to installing the Client version of the program.

There is a separate utility called Database Utility that can be installed.  Typically, this is not installed by those using their own instance of SQL Server—its purpose was to allow users to have basic maintenance and utility operations on the database (shrink, detach/attach, add users, etc).  The Database Utility does not perform any specific functions that cannot be done in traditional SQL Server tools (SQL Server Management Studio).