<p>An advantage by using Windows Authentication is that I do not need to provide a username and password to log into the server as the user currently logged into Windows. But in a specific case I would like to log into the SQL Server as another user, using the same Windows Authenticated ODBC connection.</p>

<p>So, the question: Is it possible to log into an SQL Server using Windows Authentication, but not as the user currently logged into Windows (without a dialog asking for username and password)?</p>

The simple answer to your question is no, without a dialog being presented, it's not possible to login to SQL Server using another Windows account.
The slightly more complex answer is; You can login as another user (WITH a dialog being presented) by right-clicking the Query Analyser icon and choosing "Run As" (Might require a CTRL or SHIFT key held down to make RUN AS appear.
There's also the RUNAS command line executable which you could look at, so you could create a shortcut that contains the RUNAS command, the credentials and the path to Query Analyser. That would eliminate being asked for a username or password on execution.