Thanks to vandale, here is the answer should anyone else stumble upon this question:
Use setExtendedState(Frame.MAXIMIZED_BOTH);
to initiate zoom.
But realize this is not a toggle, so clicking it again doesn't unzoom like other Mac apps. Here is the necessary code to make it a toggle:
if (yourFrame.getExtendedState() != 6)
setExtendedState(Frame.MAXIMIZED_BOTH);
else
setExtendedState(Frame.NORMAL);
Zoomed mode is state 6, normal is 0.