Cross-entropy is prefered for classification, while mean squared error is one of the best choices for regression. This comes directly from the statement of the problems itself – in classification you work with very particular set of possible output values thus MSE is badly defined (as it does not have this kind of knowledge thus penalizes errors in incompatible way). To better understand the phenomena it is good to follow and understand the relations between
- cross entropy
- logistic regression (binary cross entropy)
- linear regression (MSE)
You will notice that both can be seen as a maximum likelihood estimators, simply with different assumptions about the dependent variable.