I like the pie jaws idea, although I haven't got a good notion of how I'd locate those jaws on the mill accurately enough to mill the square pattern on center
Perhaps you could make a long 'collet' for your chuck: take a piece of round stock, machine a square hole in it (somehow) to fit over the work stock, slit the collet almost through across a diagonal drawn through the square. Align the collet in the chuck (with the slit facing inward to the chuck) so that one of the three jaws is perpendicular to the slit. This should give you a clamping action similar to a 2 jaw chuck with V jaws.
How well this will be centered is unknown, but it should be close, or within range of a few thin shims if the work stock is cold drawn or something.
First you get good, then you get fast. Then grouchiness sets in.
(Note: The opinions expressed in this post are my own and are not necessarily those of CNCzone and its management)