Winkie Country is a region in the western half of L. Frank Baum's fictional world 'The Land of Oz' from the Oz books.
They follow the Yellow Brick Road around the Emerald City to get to Winkie Country, which is just on the other side of the city. The Yellow Brick Road stops at its border, because "no one wishes to go there". 
Its people, the Winkies, were enslaved by The Wicked Witch of The West, who, as the Tin Woodman explains, rules the country mercilessly from her castle and will know you've entered her country right away. The land is dry and uneven, and completely deserted.
The Wicked Witch's castle is to the far west of Winkie country and sits on top of a cliff overlooking the West Country. It looks 'pleasant and traditional'. The only one the twins and their friends find there is a single flying monkey, Blubo. The Wicked Witch and her army have all joined the Masked Man.
From the castle, they travel into the next book, Peter Pan, and into Neverland.