Normally the evaluation measure doesn't depend on the method used, it depends primarily on the task being carried out.
Of course there are standard evaluation measures associated with broad types of tasks, such as classification or regression. There are technical constraints to take into account, for example whether the output is categorical or numerical. It's a common mistake to overlook this question and/or treat it as purely technical, but the choice of an appropriate evaluation setup should be made based on how well it represents the quality of the output of the ML process.
To answer your question: if the task is the same, the evaluation measure should be the same whether the method is linear regression or non-linear regression. Otherwise one would be measuring only some technical aspect specific to the method, not estimating the quality of the output in a comparable way.