Meaning: 好棋? (Good move?)
Yes,好棋 means "a good move"; either in chess-style games or, by analogy, anything related to a contest of strategy (e.g. 普京毁了一步好棋, "Putin screwed up a (previous) good move".)

The phrase in question, 中国好棋, is a bit different. That originated from this poster,
where it is the title of a poem:


  对弈小神童，尚在长成中。游戏无规矩，打闹是常情。但凡说故事，必须中国赢。落子成定局，欢呼轻高音。中国我爱你，年少正青春。


The character 好 has two meanings:


adjective "good"
verb "to like"


The latter meaning is the one intended in the phrase ``. So the phrase is saying "China likes chess" (or Go or generic board games). This can be inferred from the context provided by the poem, the visual emphasis on the 好 character, and also because a subject (中国) precedes it and an object 棋 follows it.

The other possible meaning for the phrase is something along the lines of a "Chinese good move". That doesn't fit the poster's context (though you can interpret it as a pun), but the phrase can be used with that meaning too. 
For instance, Chinese chess prodigy 侯逸凡 (Hou Yifan) has talked about the awakening Chinese dragon making a 中国好棋 on the chessboard of the world.



For reference, the poster: