Kawasaki disease is a medium-vessel vasculitis diagnosed by prolonged fever plus a checklist: red cracked lips and strawberry tongue, a polymorphous rash, swollen then peeling hands and feet, neck node enlargement, and red eyes.
The crucial eye detail is that the redness is a $\textit{bulbar conjunctival injection without discharge}$ - it spares the limbus and produces no pus. Therefore a $\textit{purulent}$ (discharging) conjunctivitis breaks the pattern and is the finding that does not belong; it should make you think of a bacterial or other infective cause instead.
The remaining choices - non-exudative bilateral conjunctival injection, mucosal changes, and cervical adenopathy - are all bona fide criteria.
\[\boxed{\text{Purulent conjunctivitis is NOT a feature of Kawasaki disease}}\]