I'm concerned that there's possibly several things wrong with this (namely render-square):

(defn board []
  (let [init (into [] (repeat 9 nil))
        state (r/atom init)
        handler (fn [i] (swap! state assoc i \X))
        render-square (fn [i] [:button
                               {:class-name "square"
                                :on-click handler}
                               (nth @state i)])]
    (fn [] [:div
            [:div {:class-name "status"} "Next player is: " @next-player]
            [:div {:class-name "board-row"}
             ^{:key 1} [render-square 0]
             ^{:key 2} [render-square 1]
             ^{:key 3} [render-square 2]]
            [:div {:class-name "board-row"}
             ^{:key 4} [render-square 3]
             ^{:key 5} [render-square 4]
             ^{:key 6} [render-square 5]]
            [:div {:class-name "board-row"}
             ^{:key 7} [render-square 6]
             ^{:key 8} [render-square 7]
             ^{:key 9} [render-square 8]]])))