= Solution
For $g\in\mathbb N^{\mathbb N}\cap M$ and $n\in\mathbb N$, the set
$$
D_{g,n}=\{s:\text{some }k\geq n\text{ below }|s|\text{ satisfies }s(k)=g(k)\}
$$
is dense in $\mathbb Q_0$: extend any finite sequence at one fresh coordinate with the corresponding value of $g$. The generic union $x_0$ meets every $D_{g,n}$, so it agrees infinitely often with every ground-model $g$. Thus $x_0$ is infinitely equal over $M$ and is not eventually different over $M$.
For $g\in\mathbb N^{\mathbb N}\cap M$, conditions of $\mathbb Q_1$ whose side set contains $g$ form a dense set. Once such a condition enters $G_1$, every later coordinate added to its stem must avoid $g$. Hence the generic union $x_1$ is eventually different from every ground-model $g$, and consequently is not infinitely equal over $M$.
The four answers are therefore
$$
\begin{array}{c|cc}
&\text{eventually different}&\text{infinitely equal}\\ \hline
x_0&\text{no}&\text{yes}\\
x_1&\text{yes}&\text{no}.
\end{array}
$$
Solved by gpt-5.6-sol high.
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