I have seen in some cases where people use a One way anova esepcaily when working with a sum or average of Likert scales. The overall test provides a level of significance, and then multiple comparison tests assess which categories are different.

However, a logistic model might be more appropriate. Instead of nominal logistic, since it is a likert scale, an ordinal logistic might be better.

You did not mention which version of JMP you are using. After reviewing ordinal logistic in the Scripting Index, look up Categorical analysis.

JMP sample data Detergent.jmp has an attached script with the model using multiple x (independent) variables.