There are several articles here that address this question, but none offer the help to connect-the-dots.

The attached screen shot shot shows Accessibility Inspector for the button in question, which answers to 'process java'. It also shows the properties of the window (bottom) and 2 code strings to have AppleScript press the continue button. These are:

if exists (button "Continue" of window 1) then
    click button "Continue" of window
else if exists (button "continue" of window 1) then
    click button "continue" of window 1

I am almost certain 'tab' and 'space bar' might work for this. But, can it be done through the window? If so, how?