(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.