Let me put my objection more baldly: one can postulate that one could model the behavior of the brain through a sort of reductionism, by creating elements that mimic constituent parts and elements that combine those interactions. But somewhere along the line one has to have a reasonable expectation of being able to construct the model, and I don't think it's unreasonable to insist that this expectation needs to be grounded in some successes in modelling! As it stands there's nothing standing in the way of that model having to reproduce the quantum behavior of individual molecules. And when the actual reproduction of that model could require a machine the size of the galaxy, it becomes increasingly implausible that a real model could actually be presented.
It does not follow, by the way, that non-Turing behavior implies the ability to solve non-computable problems. The real problem is that it's impossible to identify whether human behavior is or is not computable without actually computing it.
Ok...let's go back to Computation Theory 101. The definition of a computable function is a function that is Turing-Computable, that is to say a function that can be computed (or, equivalently, decided...choose your favourite term) by a Turing Machine. Therefore, by definition, any function that can not be decided by a Turing Machine is non computable.
(Oh, and if you going to insist on arguing semantics instead of focusing on the actual issues at hand, by Turing Machine I'm also referring to all Machines that are computationally equivalent to a Turing Machine, by tautology.
Now, to formalize this so our conversation can actually progress, Turing computable sets are, of course, exactly the sets at DELTA(0,1) in the arithmetical hierarchy. You claimed you could solve non-computable functions, that is those functions defined by sets that are not contained in DELTA(0,1). So I presented a problem that sits in the class of the Turing Machine's Halting problem, the Busy Beaver problem is of complexity SIGMA(0,2) in the arithmetical hierarchy, this class was a logical choice since the simple solution of it would demonstrate hypercomputational abilities on your part. Now, if you can demonstrate that you can decide all recursively enumerable sets, that would also be a valid demonstration of hypercomputational abilities as well. However, a Turing Machine is an example of a recursively enumerable set, DELTA(0,1) is contained in SIGMA(0,1). In fact, it's defined as the union of SIGMA(0,1) and PI(0,1); this union represents recursively enumerable languages in which the input is a string in the language, and a Turing Machine is capable of halting and accepting in event of such an input. However, to demonstrate hypercomputational abilities you need to demonstrate that you can halt and reject on the input of a string not contained in the language. Generally speaking, a Turing Machine will sometimes halt and reject these inputs, but other times will enter into an infinite loop. You would have to demonstrate that you can halt on (that is to say decide) ANY such input. And, to be honest, I can't even recall the class of problems in PI(0,1). But being able to show decidability in this set and obviously not just of those also contained in DELTA(0,1) would demonstrate hypercomputational abilities. If Nebel's around, he's still in school, so perhaps he can flesh out this class of computational sets for us.
There are theoretical models for hypercomputation, but they all require either infinite space or infinite time. There have been proposals for building hypercomputational computers such as computer operating in a Malament-Hogarth spacetime in the event horizon of a black hole, but this may run into theoretical problems because of our lack of our still very elementary understanding of black hole evaporation. The best bet would probably be a Superluminal Computer working on the principle of quantum entanglement, of course, for this to work you need a metamaterial with refractive index less than one, some have theorized that positrons in vacuum could accomplish this, but it hasn't been demonstrated...and, in any case, these metamaterials certainly haven't been found in the organic organ that is the human brain. Both these ideas for a hypercomputer rely on the theory of relativity to allow infinite time to pass for the computer with only finite (or no) time passing for the observer...something else that probably can't be said for you relative to everyone around you.
Computational classes are intrinsically linked to the problems that can be solved (that is to say, decided or computed) within them. You have yet to demonstrate the ability to solve any non-computable problem...no human ever has. Heck, even though there are more problems that are non-computable than computable (uncountably infinite as opposed to countably infinite), only the greatest mathematical minds in the world have managed to pose problems that can only be decided by hypercomputation, and most of these are simply problems about the behaviour of Turing Machines, though as Turing Machines we all should at least be able to do that...not only are we're Turing Machines, we seem to be rather inefficient Turing Machines.
Well, I'm going to stop there, the more I go into the formal logic behind computation theory the less sense your arguments are making. So please, enough with the rhetorical BS, focus on the mathematical logic at hand. People wonder why atheists are hostile towards theists, we're probably just tired of things like people who can't even pose a question for a hypercomputer claiming to have hypercomputational abilities and ignoring 80 years of theoretical mathematics in the process...it's frustration more than anything else.