If you recast the paradox as "There exists a machine scientists made that will determine if any other intelligence will either choose the box or not the choose the box. You are that other intelligence. Do you choose the box or not choose the box?" then you can see that the same principle holds; if you incorporate the logic of the predictor into your own intelligence, you can twist the original machine's logic back on to itself in exactly the same way the Halting problem does.
A subtlety to note about the halting problem is that while it is phrased in terms of that particular machine that can twist back on itself, it is itself a generalized proof of impossibility, and via Rice's theorem and lots of other work over the years it extends out into the impossibility of all sorts of other machines as well. The proof simply provides one machine that unambiguously can not be created, it is not limited to that one machine.
Similarly, while a human brain that encompasses the same logic as the box-prediction machine may have technical problems in the fuzzy real-world land, the fact that such a brain would be fundamentally unpredictable means that the prediction machine can't exist.
If you try to fix up the thought experiment by limiting the size of the predictor and the predictee in various ways, I suspect it isn't that hard to show that the predictor must be exponentially more complicated than the predictee in order to function, and something "exponentially larger" than the human brain doesn't really fit in the universe. And then if you try to escape by allowing arbitrarily large mathematical systems, you're back where I described above. If you try to bound how much larger the predictor has to be than the predictee, you are going to be encountering some really serious mathematical problems doing so (of the "busy beaver" mathematical sequence sorts).
Given that the problem intrinsically encompasses a delay between the scan and the decision, I can simply take an ad-hoc hash function of the experiences of the last day and now the predictor needs to have also had information of my entire last day as well, and a simulation sufficiently detailed to have predicted that, too. Even if it can predict the hash function I would use (itself no guarantee since that is also conditionalized on the intervening day, potentially), it can't predict the input going into it.
I think most people intuitively sense that the predictor can't really exist; I think people's intuitions are correct. It would require the predictor to have physically impossible amounts of additional resources vs. the predictee if you try to embed it in real space and time, and if you put it in math space it's nothing more and nothing less than the halting problem restated.