Non - aromatic nature of quinones
The reason you must be able to draw double bonds within the ring is the $4n+2$ rule applies rigorously only when the conjugated atoms are all in such a ring. If you are drawing double bonds to adjacent atoms off the ring then you don't have all the conjugated pi electrons in the ring anymore, and then there is no longer a $4n+2$ rule for aromaticity or a $4n$ rule for antiaromaticity.

In some systems it is possible to render contributing structures that have all the conjugated electrons in the ring and meet the $4n+2$ rule, in which case you can assert that this contribution is enhanced by aromaticity and the properties of the compound affected accordingly.  For example, the aromatic contribution to cyclopropenones leads to enhanced polarity of the carbonyl group and increased basic character of the oxygen atom, compared with ordinary ketones.



Illustration source: https://commons.m.wikimedia.org/wiki/File:C3H2OResonance.png.  Drawn by WP user Smokefoot.  Link to License

Quinones, however, cannot be given such a structure without leaving the highly electronegative oxygen short of an octet, so they lack any significant aromatic contribution.