Where is the hypothesis that $X$ and $Y$ are path-connected being used? 
The above is from Hatcher's book.
I am curious exactly where is the hypothesis that $X$ and $Y$ are path-connected being used in the proof? I don't see it being invoked explicitly, nor can I see where it is implicitly used.
Also, a second question would be are there any counter-examples where the proposition does not hold if $X$ or $Y$ are not path-connected?
Thanks!

There is confusion in the literature between a space and a space with base point. Grothendieck wrote to me in 1983:"... the choice of a base point, and the $0$-connectedness assumption, however innocuous they
may seem at ﬁrst sight, seem to me of a very essential nature.  To make an analogy, it would
be just impossible to work at ease with algebraic varieties, say, if sticking from the outset (as
had been customary for a long time) to varieties which are supposed to be connected.  Fixing
one point, in this respect (which wouldn’t have occurred in the context of algebraic geometry)
looks still worse, as far as limiting elbow-freedom goes!...." He was also keen on the use of the fundamental groupoid $\pi(X,C)$, or $\pi_1(X,C)$,  on a set $C$ of base points, chosen according to the geometry of a situation.  
The particular issue of products is dealt with in Topology and Groupoids by showing that the natural morphism  $\pi(X \times Y ) \to \pi(X) \times \pi(Y)$ is an isomorphism, where $\pi(X)$ denotes the fundamental groupoid.  What do you expect for $\pi(X \times Y, C\times D)$? 
The general issue of  using groupoid (and higher groupoid)  structures  in algebraic topology is discussed as part of  this preprint Modelling and Computing Homotopy Types:I. In particular, $\pi_1(X,C)$ is useful for modelling and computing homotopy $1$-types. 
