Well, I guess you could find a piece of paper with additional information in the visited square. Somewhere the extra information must come from, after all.

As you can see, the probabilities $P(F)$ do get updated post a null search result, but it doesn't provide clues as to which of the next squares to search.

Of course it does. If the top right square now has a probability of $2/3$, then this means it is twice as likely in the upper right square than not. Which is a very strong hint that this square is the one which should be looked at next. Indeed, the probability distribution is exactly that: The information we have about the square in which we will find the object.

Originally, the probability is $1/4$ in each square, which tells us that we know the object is in one of the squares (because the probabilities for the squares add up to 1; if they didn't, there would have to be a fifth option with positive probability, because the sum of all options must be one), but we have zero information about which one it is in (because the probability is equal in all squares).

Searching that square tells us that the object is not there, but gives us zero information in which of the other squares it will be found, therefore the three other squares still have equal probability.