Aromaticity, and its antonym too, are the properties of simple cyclic systems of alternating single and double bonds. The use of the term elsewhere is an abuse of language.
That said, in practice people tend to apply the word "aromatic" to all kinds of systems. One just has to remember, though, that it is not a clearcut black-or-white characteristic and not a precise measure of anything. King Richard I of England was said to have the heart of a lion. That's a metaphor. When someone calls, say, naphthalene aromatic, it is the same thing.
Now to the point: is that non-rigorous use of "aromatic" justified when applied to cycles with side chains? The answer is: yes or no, depending on the particular system. You may look up the bond length, or compare the stability via enthalpies, or (if neither of these is available) calculate the orbital energies and see how it all turns out. Indeed, squaric acid is not really aromatic in that vague sense, but its dianion definitely is. So are the dianions of its triangular, pentagonal, and hexagonal siblings, known as deltic, croconic, and rhodizonic acids. On the other hand, 1,4-benzoquinone does not look aromatic at all.
Nature is under no obligation to give us a rule of thumb for deciding aromaticity just by looking at the formula. That such rule exists for simple cycles is a rare and lucky coincidence.
So it goes.
