(From SGA I, Exposé XI).
You can prove it using the following two facts:
A product of simply connected proper varieties is simply connected (SGA I, X, 1.7). (*)
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.
