Probably this question can be phrased in a much greater generality, but I will just state it in the generality I require. I work over $\mathbb{C}$.



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> Let $C\_1, C\_2 \subset \mathbb{P}^1$ be non-empty open subsets and $f: X \to C\_1 \times C\_2$ a non-trivial finite etale cover. Does there exist $i\in \{1,2\}$ such that the composition $X \to C\_1 \times C\_2 \to C\_i$ has non-connected fibres?
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Finite etale covers of products of curves
The answer is *no*, at least in general, as shown by the following counterexample.


Take a double cover $\bar{f} \colon \bar{X} \to \mathbb{P}^1 \times \mathbb{P}^1$, branched over a reducible 
curve of the form $B=L\_1 + L\_2 + M\_1 + M\_2$ (here $|L|$ and $|M|$ are the two pencil of lines on the quadric). 


Such a cover exists because $B$ is $2$-divisible in $\mathrm{Pic}(\mathbb{P}^1 \times \mathbb{P}^1)$, and corresponds to an étale cover $f \colon X \to C\_1 \times C\_2$, where each $C\_i$ is $\mathbb{P}^1$ - {two points}. 


If these points are (say) $0$ and $1$ in both factors, then the equation for $X \subset \mathbb{C} \times (\mathbb{C}-\{0, \, 1\})^2$ is  

$$z^2 = xy(x-1)(y-1), \quad f(z, (x, \,y)) = (x,\, y).$$


It is clear that the general line in $|L|$ and $|M|$ intersects the branch locus $B$ transversally at two points, hence both compositions $$\bar{X} \to \mathbb{P}^1 \times \mathbb{P}^1 \to \mathbb{P}^1$$ have connected fibres, the general one being isomorphic to $\mathbb{P}^1$ (double cover of $\mathbb{P}^1$ branched at two points). 


Then both compositions $$X \to C\_1 \times C\_2 \to C\_i$$ have connected fibres, the general one being isomorphic to the $\mathbb{P}^1$ above minus the ramification, i.e. $\mathbb{P}^1$ minus two points, that is clearly connected.


In the same vein, choosing as $B \subset \mathbb{P}^1 \times \mathbb{P}^1$ a divisor of type $$B = \sum\_{i=1}^{2g+2} L\_i + \sum\_{i=1}^{2g+2} M\_i,$$
both compositions $$\bar{X} \to \mathbb{P}^1 \times \mathbb{P}^1 \to \mathbb{P}^1$$ have connected fibres, the general one being isomorphic to a hyperelliptic curve $\Sigma\_g$ of genus $g$, and so both compositions $$X \to C\_1 \times C\_2 \to C\_i$$ (here each $C\_i$ is $\mathbb{P}^1$ minus $2g+2$ points) have connected fibres, the general one being isomorphic to $\Sigma\_g$ minus $2g+2$ distinct points. 


