Do I have to standardize my new polynomial features?
It's better to do the square computing on the values of the non standardized features, and then standardize these new features. 

The point of standardizing features is to put features in a similar scale, e.g. 0-1. If you didn't standardize these new features after the computing, although your new features are between 0-1, your new features will be one or more magnitude smaller than your old features.