I'd not suggest to hardcode the connection string as this ties your application to a very specific deployment scenario. What if a customer wants to use his or her existing SQL Server instance? What if the customer creates a named instance for the SQL Server Express installation (e.g. SERVERNAME\SQLEXPRESS). Sooner or later you'd have to change your code to reflect the different situations and you'd do the same thing that you can do already now.
In .NET, you can store a connection string in an application configuration file. It is named as the Exe file but has a .config extension (e.g. MyExeName.exe.config). In Visual Studio, all you need to do is to add a file to your project ("Application Configuration File", app.config) if it doesn't exist yet. Upon build, it will be renamed to the Exe-name plus .config. See this link for detailed information.
For your scenario, you'd have a configuration file like this. It references the local computer by using "." as the server name. So you don't have to change it after deployment if this is your default scenario:
<?xml version="1.0" encoding="utf-8" ?>
<configuration>
<connectionStrings>
<add name="MyConnection"
connectionString="Data Source=.;Initial Catalog=myDatabase;Integrated Security=SSPI"/>
</connectionStrings>
</configuration>
In your code, you can access the connection string using the ConfigurationManager class (you might need to add a reference to the System.Configuration assembly). You can use the acquired connection string when you create the connection:
Dim connStr = ConfigurationManager.ConnectionStrings("MyConnection").ConnectionString
Using conn = new SqlConnection(connStr)
conn.Open()
' Use the connection
End Using
To summarize: you'd add a default configuration file with a connection string that works for any server name to your deployment. If the customer wants to have the database on a different instance or another server, you simply change the connection string - but you don't have to if the customer is fine with the default configuration. As you see, it is not much more effort than to hard code it, but it will make your deployments more flexible.