Extended game mixed SPNE
There are a couple of imprecisions. First, there are 3 subgames: two that start with player 2 moving, and the complete game is also a subgame. 

Second I think that A is a best response for player 1 iff $r\geq \frac{z-y}{x-y}$ otherwise, B is the best response (perhaps this was just a typo). 

Third, note that the fraction is guaranteed to be positive and smaller than 1 given the restrictions on the parameters,  but it is not necessarily equal to $0.5$, For example, let $x=5$, $z=3$ and $y=2$, then the fraction is equal to 1/3. 

Fourth, you are missing other equilibria, when $r=\frac{z-y}{x-y}$, then player 1 is indifferent between A and B, and any randomization of A and B also constitutes an equilibrium.

Lastly, when writing the equilibria, you must specify what player 2 does in each information set (or node in this game) 

Thus: $$NE=\{(A, (ra+(1-r)b)b)|r\geq \frac{z-y}{x-y}\}\cup\{(B, (ra+(1-r)b)b)|r\leq \frac{z-y}{x-y}\}$$
$$\cup \{((pA+(1-p)B), (ra+(1-r)b)b)|r=\frac{z-y}{x-y}, 0\leq p\leq 1\}$$

Where $(ra+(1-r)b)b$ specifies that player 2 randomizes in the first node and chooses b in the second node.