Different neural structures for quick vs. methodical decision making In the book "Thinking, Fast and Slow", the author claims that there are two systems of thinking. System 1 is quick, instinctive and emotionally driven while System 2 is more logical and deliberate.

Firstly, has there ever been any physiological proof, in terms of brain activation areas or other signs, that these two systems exist. If they do exist, have they ever been capture in a cognitive model, regardless of biological plausibility?
I would disagree with @Krysta: the distinction between system 1 and system 2 processes goes far beyond that of cognition with/without emotion: it's a complex debates that's been going on since the 70s (in it's current form, it's echoed in a wider debate going back centuries). For the sake of brevity, let me resort to bullet points.


For psychologists, Evans (2008) is the go to reference on this matter. Obviously, Kahneman is the grandfather of this topic, and his book is a great introduction for the layman, you'll find a much more rigorous discussion here.
There's a lot of variation in terminology here - while "dual process theory" is the commonly accepted generic term for this idea, you'll also see it as "type, or system 1/2 processes", "intuition/logic", "associative/controlled processes", etc. Again, see Evans (2008).
A nice simplification of the distinction would be to say there's two kinds of processes, System 2 processes, which requires Working Memory (i.e. maintaining, updating, and manipulating a representation of information over a period of time), and System 1 processes, which don't. System 1, when taken this way, isn't a single "subsystem", its a diverse set of processes (this idea is nicely laid out in Stanovich, 2009,
who uses the term "The Set of Autonomous Systems (TASS")),
amongst other sources) including evolved, domain specific modules for solving specific problems (i.e. recognising faces), the emotion subsystem, and rules and associations which have become automatic through experience, and implicit learning (stereotypes, for example).


To finally get to your question, yes, this does translate into specific neural systems.


Working Memory (and thus system 2 processes) is pretty clearly localised to the prefrontal cortex (see Braver et al, 1997, for example).
System 1, not being a single "system", obviously can't be localized to a single brain region. However, many of it's component processes can be: emotion to the limbic system, face recognition to the fusiform face area, etc. (I don't know much about localization of implicit learning, but I would guess the hippocampus is involved).





  I've just realised I haven't answered your second question, about cognitive models. I'll come back to this at a later date.
  EDIT I've just noticed you're working with Chris Eliasmith. I can think of few people better qualified to answer your second question than he is.




Reading

Evans, J. S. B. (2008). Dual-processing accounts of reasoning, judgment, and social cognition. Annu. Rev. Psychol., 59, 255-278.

Stanovich, K. E. (2009). Distinguishing the reflective, algorithmic, and autonomous minds: Is it time for a tri-process theory. In two minds: Dual processes and beyond, 55-88.