Named Credentials causing System.UnexpectedException: Script-thrown exception on HTTP callout error

There could be multiple reasons for the above error message. Before raising a case with support, it would be best to re-verify the below steps to debug further. 

1. Verify if the callout name used in Apex Methods matches the Named Credential’s API name under Setup-> Named Credentials. 

If there is a mis-match, kindly make sure to change and use the same syntax/api name of the Named Credentials.
Ref:  Named Credentials as Callout Endpoints

2. Verify if the URL has any path that needs to escape URL reserved characters 

3. Verify if the remote endpoint has any stricter URL path filtering that may have been enabled. Example: Replacing “\” with “%5c”. This is controlled at the endpoint level and not at salesforce.

Additional Resource: Test classes fail with error 'System.SObjectException: Script-thrown exception'