The example strikes me as a level specifically designed to look like a boolean equation, while the other levels in the app website are quite different.
It doesn't make sense to consider any particular level as NP-complete or not, much like it doesn't make sense to consider any particular boolean proposition as NP-complete or not. Even a particularly complicated level or a particularly complicated formula has a trivial program solving it, one that simply prints the solution to that particular puzzle/level without doing any real computation at all.
But in the article is said: The proof is surprisingly simple and he did it by showing how the boolean satisfiability problem (SAT) can be mapped onto a level in the game.
And only an example level was given, one that looks like it was designed for that purpose. What I can't understand is how a "real" level of the game (like this one: http://bit.ly/bOWbgi ) can be mapped in a boolean equation.
The proof of NP-hardness is basically
For every instance of a SAT problem, there exists some puzzle whose solution encodes the SAT solution.
You are asking about
For every instance of his puzzle, does there exist a SAT formula solved by this puzzle
Which is a different question.
Specifically: A problem X is NP Complete if:
1: X is in NP.
2: Every problem in P can be reduced to X in polynomial time.
We have established the first condition by intuition-- pathfinding is easy. Immediately, we know, then, that there exists some way to reduce this to 3-SAT, in a way general for any level. Reducing 3-SAT to this, however, proves the second criterion (because 3-SAT is NP-complete, so by doing two reductions you can reduce any NP problem to this, in time at worst the cost of doing sum of the two polynomials describing the complexity of the reductions).
So we can also convert 3-SAT-- or any NP-complete problem, in fact, and even more generally, any NP problem-- to this in poly-time.
A rather incredible theorem is the Cook-Levin theorem which states that any problem that can be verified in polynomial time can reduced to an instance of sat SAT (in polynomial time).
The practical upshot of this is that to show NP-completeness, you only need to provide a polynomial certificate and reduction to SAT, or some other know NP complete problem.