neg_mean_squared_error for optimizing classifier

In the notebook examples below, there is a cross validation step using neg_mean_squared_error for both first stage Regressor and classifiers. My understanding of this is that we want to minimize the mean squared error for the final model (as below).

The approach allows for arbitrary Machine Learning algorithms to be used for the two predictive tasks, while maintaining many favorable statistical properties related to the final model (e.g. small mean squared error, asymptotic normality, construction of confidence intervals). https://econml.azurewebsites.net/spec/estimation/dml.html

However, what is the implication of using a regression evaluation metric for classification cross validation?

https://github.com/microsoft/EconML/blob/master/notebooks/AutomatedML/Automated%20Machine%20Learning%20For%20EconML.ipynb
https://github.com/microsoft/EconML/blob/master/notebooks/ForestLearners%20Basic%20Example.ipynb

Start by comparing the cross-validation configurations in AutomatedML/Automated Machine Learning For EconML.ipynb and ForestLearners Basic Example.ipynb, focusing on classifier uses of neg_mean_squared_error. Read the linked DML specification and determine whether the examples use an appropriate classification evaluation metric; done means the behavior and its implication are clearly resolved for both notebooks.