When is the generic fibre geometrically irreducible?

Question

Let $X \to Y$ be a morphism of schemes. According to Exercise 6.15 in Görtz—Wedhorn, if Y is integral with generic point $\eta$, then the generic fibre $X_{\eta}$ is irreducible provided that $X$ is irreducible. My questions are as follows:

1. Can someone please supply a proof of the above fact?
2. If X is a scheme over $\mathbf{Q}$ that is geometrically irreducible, would it follow from the above exercise that $X_{\eta}$ is also geometrically irreducible?

Thanks for your help.