Art of Electronics: Emitter-Follower Zout
The standard way of doing this is to use small-signal AC analysis. Assume the transistor is biased in the forward-active region. Use the hybrid-pi model. Then place a test voltage/current source at the output node and ground the input. Measure the current/voltage of your test source and that tells you the output impedance. You can also find the input impedance that way.

This is basically the same as what the book is telling you to do, except that using the small signal model of the BJT allows you to turn problem into a linear circuit analysis problem which should be easy to do mechanically.

I'm not sure what is wrong with your derivation but the 0.6V should somehow drop out because you're looking at the change in voltages and currents.