Excel Data Connection to MS-SQL using Windows Auth from untrusted machine

We have an Excel Spreadsheet that uses a Data Connection to a MS-SQL database. The database server is joined to our domain, and Windows Authentication is enabled.

Here is a sample of the connection string (with revealing info obfuscated):

Description=DataConnection;DRIVER=SQL Server;SERVER=10.x.y.z;UID=domain_user;Trusted_Connection=Yes;APP=Microsoft Office 2010;DATABASE=DatabaseName

And the Authentication Settings are set to "Windows Authentication"

This works perfectly fine from a domain-joined / trusted machine. It doesn't even ask for credentials; it just works flawlessly when I hit Refresh.

It will work from a non-domain / un-trusted workstation, if I use SQL Authentication. But we are trying to move away from that.

Likewise, it will from a non-domain machine using Windows Authentication if I use "runas" to launch Excel with domain credentials, as follows:

"runas /netonly /user:domain\user %path-to-excel%\excel.exe"

However, using "runas" is a kludge, and our employees will almost certainly have issues... even if I were to create batch files to help automate it.

When launching Excel normally (without using "runas") from a non-domain machine, it gives this error when I hit Refresh:

Upon hitting OK, I have the option to un-check "Use Trusted Connection" and manually enter a user/password. I put in a valid user that would be able to connect fine from a domain-joined machine, or that worked fine using "runas". This is the error it gives with that:

At this point, I have not a clue what to do to get Excel to successfully connect with domain credentials from a non-domain machine.

Ideas?