Can game theory be used to predict economic specialization (division of labor)? There is a hint of a connection to game theory in Adam Smith's Wealth of Nations. For example, he says: 


  It is not from the benevolence of the butcher, the brewer, or the
  baker that we expect our dinner, but from their regard to their own
  interest.


We can put his general argument into a payoff matrix, like shown below:



There are two Nash Equilibria here: {DS,DS} and {S,S}.

Obviously, if they both specialize, they are both much better off. However, each of them not specializing is also an equilibrium.

Is there any way that Game Theory predicts the emergence of specialization in a population of such players?
I think you're over-analyzing here. Game theory is not just about maximizing your own pay-off, but about doing the best you can given the behavior of others. What Smith describes is just that people do what they're good at to make a living. That is, the way I read it, Smith claims that people specialize independent of what others do.

That being said if the emergence of specialization was to be predicted with game theory I'd look into evolutionary game theory. I'm not sure if you'll find anything though.