= Solution
Every move can be drawn from a <full conditional distribution>, producing a <Gibbs sampler> with acceptance probability one. Write $s_{x,i}^2=\sigma_{x,i}^2$ and $s_{y,i}^2=\sigma_{y,i}^2$. First update independently
$$
\xi_i\mid-\sim N\!\left(
V_{\xi i}\left[\frac\mu{\tau^2}+\frac{\beta(\eta_i-\alpha)}{\sigma^2}+\frac{x_i}{s_{x,i}^2}\right],V_{\xi i}\right),
\quad
V_{\xi i}^{-1}=\frac1{\tau^2}+\frac{\beta^2}{\sigma^2}+\frac1{s_{x,i}^2},
$$
and then
$$
\eta_i\mid-\sim N\!\left(
V_{\eta i}\left[\frac{\alpha+\beta\xi_i}{\sigma^2}+\frac{y_i}{s_{y,i}^2}\right],V_{\eta i}\right),
\quad
V_{\eta i}^{-1}=\frac1{\sigma^2}+\frac1{s_{y,i}^2}.
$$
Let $X$ have rows $(1,\xi_i)$ and $\eta=(\eta_i)$. Update the <linear regression> coefficients jointly by
$$
\begin{pmatrix}\alpha\\\beta\end{pmatrix}\Bigm|-
\sim N_2\!\left((X^TX)^{-1}X^T\eta,
\sigma^2(X^TX)^{-1}\right),
$$
and update
$$
\mu\mid-\sim N\!\left(\bar\xi,\frac{\tau^2}{N}\right).
$$
Finally, the flat positive variance priors give the following full conditionals, each an <inverse-gamma distribution>:
$$
\sigma^2\mid-\sim\operatorname{InvGamma}\!\left(\frac N2-1,
\frac12\sum_i(\eta_i-\alpha-\beta\xi_i)^2\right),
$$
$$
\tau^2\mid-\sim\operatorname{InvGamma}\!\left(\frac N2-1,
\frac12\sum_i(\xi_i-\mu)^2\right).
$$
A systematic sweep in the displayed order, using the newly sampled values immediately, defines the chain. The shapes are positive for $N>2$; full column rank of $X$ is also required.
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