If you are trying to use integrated security, your connection string should be:
"Server=localhost;Database=greenapplication;Trusted_Connection=True"
You then manage permissions through roles and windows groups. (You can grant individual users permissions, but this is not the recommended way to manage database access).
UPDATE: You neglected to mention in original question that this is a web application. The usual way to handle this is to create an app. pool running under a known identity (say NETWORK SERVICE), and then give that identity the necessary permissions on the database.
Here's an example:
-- Create a SQL Server login for the Network Service account
sp_grantlogin 'NT AUTHORITY\Network Service'
-- Grant the login access to database
USE MyDB
GO
sp_grantdbaccess 'NT AUTHORITY\Network Service', 'Network Service'
-- Add user to read only database role
USE MyDB
GO
sp_addrolemember 'db_datareader', 'Network Service'
If you insist on using the less secure method of passing of username and password in connection string, use SQL Server logon: