Catalysis of the reaction: (3S,4R)-3,4-dihydroxycyclohexa-1,5-diene-1,4-dicarboxylate + NAD+ = 3,4-dihydroxybenzoate + CO2 + NADH.
cis-4,5-dihydroxycyclohexa-1(6),2-diene-1,4-dicarboxylate:NAD+ oxidoreductase (decarboxylating)