(From SGA I, Exposé XI).

You can prove it using the following two facts

1. A product of simply connected proper varieties is simply connected (SGA I, X, 1.7). (*)

2. The fundamental group, in particular, being simply connected, is a birational invariant of proper regular varieties (SGA I, X, 3.4).

(*) I do not know whether the properness is necessary here; it is required for the more general computation of the fundamental group of a product: in positive characteristic, one of the factor needs to be proper.

(From SGA I, Exposé XI).

You can prove it using the following two facts:

1. A product of simply connected proper varieties is simply connected (SGA I, X, 1.7). (*)

2. The fundamental group -- so in particular, being simply connected -- is a birational invariant of proper regular varieties (SGA I, X, 3.4).

(*) I do not know whether the properness is necessary here; it is required for the more general computation of the fundamental group of a product: in positive characteristic, one of the factors needs to be proper.