First off, you shouldn't have to "burn a copy" specifically for a connection string - that should be in a configuration file so your app does not need to change, and be set at installation time (with a way to change it if the details change in your app) when you check or create the DB for the first time.
The user should know - or be able to find out - what the connection string detail for his instance of SQL Server / MySQL from his DBA, though it is possible to "help" him by locating the SQL server instances:
SqlDataSourceEnumerator instance = SqlDataSourceEnumerator.Instance;
DataTable dt = instance.GetDataSources();
foreach (System.Data.DataRow row in dt.Rows)
{
Console.WriteLine("ServerName = {0}", row["ServerName"]);
}
You can't - or shouldn't - know the login details for his DB: it includes a password for a reason!