I just had a customer ask me this same question - overall in JMP we use Westguard and Western Electric rules and circle the points rather than use general p-values like Minitab. Test info is further down in this post.
Control charts can be done in the Control Chart Builder (Analyze -> Quality and Process -> Control Chart Builder). This is an interactive drag and drop control charting platform. There is a control panel on the left that will let us control different aspects of what we see, for example individual points versus the mean per x grouping variable.
JMP uses Western Electric/Nelson rules (Test 1-8), testing beyond limits, and Westguard rules - the points that trigger these alerts are circled. Tests are described fairly well. So rather than getting just a p-vlaue like in Minitab, you will see the points circled with the test beside it.