I dare say that for me a point is both a horizontal and a vertical segment... Perhaps you are right that this isn't perfectly obvious, but I can't see any definition of a horizontal/vertical closed segment that would assume otherwise (unless you choose to specify explicitly that a segment must have positive length?).
Another thing, the area of a rectangle is a real number, not an integer (only the particular value we are looking for happens to be an integer). The infimum of the set of possible areas is most certainly 0, so no answer other than 0 can ever be considered. The question whether the word 'smallest' is legal in this context depends on the definition of a segment. If you think it is objectionable I can change it to infimum .