To run any executable including Chrome in JAVA:
If the path to the application is a system variable:
String location = System.getenv("APPVARIAVLE");
Process process = new ProcessBuilder(location).start();
Or if you want to use the fully qualified path:
Process process = new ProcessBuilder("C:\\location\\MyApp.exe").start();
The JavaDoc for the process builder say that you can add parameters like this:
new ProcessBuilder("myCommand", "myArg1", "myArg2");
The argument for incognito looks like it is: "-incognito" and to open a url just add the url: "example.com".
Which means that you can most likely can add the url and incognito arguments the following way to chrome in the arguments:
Process process = new ProcessBuilder("C:\\YourChrome\\Location\\chrome.exe","-incognito","http://stackoverflow.com").start();