Your Authentication Provider configuration is corrupted. The relevant source:

(IUnicornAuthenticationProvider)Factory.CreateObject("/sitecore/unicorn/authenticationProvider", false);

It sounds like the Sitecore Factory is loading up a string instead of a type. Using /sitecore/admin/showconfig.aspx verify that the authenticationProvider element has a valid type attribute.

Verify also that your configuration matches the Unicorn version you have installed (which was not specified), and that the PowerShell script you're using is likewise from the same Unicorn version (releases are tagged on GitHub).

You should also be able to provoke the issue by hitting /unicorn.aspx?verb=Challenge - in a correctly configured environment it will show a GUID.

This is caused by having the Unicorn.SharedSecret.config loading prior to Unicorn.UI.config (which prevents the type definition on the auth provider from loading correctly due to Sitecore config patching rules).

Rename your Unicorn.SharedSecret.config to Unicorn.zSharedSecret.config and that should fix it.