So I agree youre getting oxidation, and the reason is that hydroquinone, when deprotonated, is converted to benzoquinone. Benzoquinone is known for undergoing all sorts of oligomerization events. I know this is a strange suggestion, but perhaps try doing this with O2 present. Some of these side reactions are radical in nature and will be quenched by oxygen. I do not think a little water will hurt your reaction much if you leave it open to air.
I suggest checking scifinder before trying again on this one, I'm sure people have tried to double-Williamson ether hydroquonine before.