I know nothing about how SPSS works, but it sure looks like you fit a good old fashioned ordinary least squares regression model with 0 and 1 as continuous responses, saved the studentized residuals and then did a scatter plot of each set of residuals by the levels of the response. If you set up a data table in JMP you can replicate the same erroneous way to model a categorical response. Far be it from me to tell you your professor was wrong...but...
I strongly recommend you follow the advice of Karen's earlier response. If you are going to use JMP you should define the response in the JMP data table as a character data type, 0 and 1 are acceptable, but make sure you articulate the data type as character, and the JMP modeling type as nominal. An alternative way to characterize the response is to use some phrase that may have more practical meaning than 0's and 1's. Maybe use JMP's Column Recode capability to recode the 0 to something like 'negative' and the 1 to something like 'positive'. I have found this technique useful when communicating results because then you never have to remember what the 0 and 1 stand for from a practical point of view.
Then JMP will guide you to make sure in the Fit Model platform when you choose Nominal Logistic Regression as the modeling personality. Essentially if you define the column properties correctly in JMP...JMP mistake proofs you away from executing an OLS model for this problem.