I struggled for a while and here is the answer for anyone else who is in need.
Hope this helps.

OK The book and help are pretty good but I had to really research this to find the answer to:
How can I have an ADO connection connect to a remote server using the username and password
of the db?

My Server is called QA-D
The DB Server I'm connecting to is QA-D\QA_SQLExpress
My db is Adventureworks (since we can all get that)
I'm using UserQA-D2K5/sapassword who is mapped as an owner of this db.

What I had to do is comment out the "Integrated Security" string.
Then for the username the value to set is "uid".
For the password the value to set it "MyPassword".

This code below connects to my server and retrieved the data for me.
var AConnection, RecSet, Cmd, Prm, MyValue;
// Create a new Connection object
AConnection = ADO.CreateADOConnection();
AConnection.ConnectionString =
"Provider=SQLOLEDB.1;" +//Integrated Security=SSPI;" +
"Persist Security Info=False;" +
"uid=UserQA-D2K5;password=MyPassword;" +
"Initial Catalog=AdventureWorks;Data Source=QA-D\\QA_SQLEXPRESS";

// Activate the connection

// Execute a simple query
RecSet = AConnection.Execute_("SELECT Title FROM Employee WHERE EmployeeID > 286");

// Iterate through query results and insert data into the test log
while(! RecSet.EOF)