For now at least, we don't know of any counter-example to the Church-Turing thesis. Humans are certainly not able to tell whether a program in general will halt for any input it recieves. If you believe otherwise, please tell me whether the following program halts:
Foo():
if P = NP:
return true
else:
Foo()That's a logical fallacy - just because I can't show whether P=NP doesn't mean nobody can!
Turing-Church is a conjecture, not a theorem. There have been arguments made against it [1].
Also, the link you replied provides an argument of efficiency, not an argument of possibility. It is arguing that we might use things that are not Turing machines because they can solve the problem more efficiently, not because they can solve problems Turing machines could never hope to solve.
If there is a single Turing machine for which there is no human that can tell whether it will halt, the conjecture that human reasoning is not more powerful than Turing machines is still not disproven.
Also, if you were right that humans can do things that Turing machines can't, it would be an interesting excersise to try to find that example of super-Turing human reasoning and try to see why it can't be encoded as a Turing machine. No one has successfully done this yet, so there is no reason to believe that it is possible.
And yes, humans are very well capable of solving that, because humans can in fact proof whether any program halts.
This doesn't mean every human can or that it has been done for every program, but it means that humans are - in general - able capable of deciding this, while algorithms (programs) can't.