Assuming you're starting with square stock, I'd try this. When standing in front of the lathe, with the work spinning toward you, as with a conventional lathe, plunge the router straight down from the top of the piece, on the RIGHT side of thefinished square section, like a parting tool. Plunge very slowly to prevent tearout. Having the tool slightly off center toward the front or rear of the piece might help if your seeing tearout. Trial and error will be your friend.
To do the left side of the square section, reverse the rotation of the spindle so that the cutting action is the same.
Increase speeds until you see tearout, to find the limits. I'd also try with opposite rotation and a climb cut to see which works better.
I think coming in from the side might cause more tearout, because the depth of cut would be greater, and you'd be cutting endgrain in light passes, which almost always tears out, especially with oak.
One last thing. Always use very sharp tools.