The regression specification here is actually
$$y_i =\beta_1x_i + \nu\cdot I_{\{i=n+1\}} +\epsilon_i, \;\;i=1,...,n+1$$
which leads us to the OLS estimator
$$\begin{pmatrix}
\hat \beta_1\\
\hat v
\end{pmatrix} =
\frac{1}{\sum\limits_{i=1}^{n+1} x_i^2 - x_{n+1}^2} \begin{pmatrix} 1 & -x_{n+1} \\ -x_{n+1} & \sum_{i=1}^{n+1}x_i^2 \end{pmatrix}
\begin{pmatrix} \sum_{i=1}^{n+1}x_iy_i \\
y_{n+1} \end{pmatrix}
$$
where the last $2\times 1$ vector is $Z'y$. So we obtain the following equation for the OLS estimator of $\beta$:
$$\hat \beta_{1,n+1} = \frac{1}{\sum\limits_{i=1}^{n+1} x_i^2 - x_{n+1}^2} \left(1\cdot \sum_{i=1}^{n+1}x_iy_i - x_{n+1}y_{n+1}\right)$$
Note that $$\sum\limits_{i=1}^{n+1} x_i^2 - x_{n+1}^2 = \sum\limits_{i=1}^{n} x_i^2$$
and
$$1\cdot \sum_{i=1}^{n+1}x_iy_i - x_{n+1}y_{n+1} = \sum_{i=1}^{n}x_iy_i$$
So
$$\hat \beta_{1,n+1} =\frac {\sum_{i=1}^{n}x_iy_i}{\sum\limits_{i=1}^{n} x_i^2} = \hat \beta_{1,n}$$
i.e. it is the same as the estimator we obtain from the sample of size $n$.
This generalizes to the case of many regressors, and of intermediate observations, and leads to the following general conclusion: in OLS estimation, including a dummy variable that takes the value $1$ for only one observation, and is zero in all others, effectively removes the observation from the sample, as regards the estimation of the coefficients of the other regressors. This is the classic way to treat "outliers" or an observation where "something extraordinary has happened", and one feels that this observation would misleadingly affect the estimation of the parameters, estimation which attempts to capture some "long-term", "structural" relation between the dependent variable and the regressors.