The question says it all:
Given two maps $f\colon A\rightarrow B$ and $f'\colon A'\rightarrow B'$, such that their product $$f\times f'\colon A\times A'\rightarrow B\times B'$$ is a homotopy equivalence, must $f$ and $f'$ be homotopy equivalences?
This certainly holds for weak equivalences and therefore for spaces of the homotopy type of CW complexes, but what about the general situation?
Bonus question: What about the case of infinite products?
I think, in general situation $f$ and $f'$ will be homotopy equivalences too.
Denote $f\times f':A\times A'\to B\times B'$ by $F$ and let $G:B\times B'\to A\times A'$ be the map such that $G\circ F\sim Id_{A\times A'}$ and $F\circ G\sim Id_{B\times B'}$. Choose one point $a'\in A'$. Then let $g:B\to A$ be the composition $p\circ G\circ \iota_B:B\times f'(a')\to B\times B'\to A\times A'\to A$, where $p$ is projection of a product and $\iota_B$ is just inclusion.
Let's check that $g\circ f\sim Id_A$. Note that $g\circ f$ can be written as the composition $p\circ G\circ F\circ \iota_A:A\times a'\to A\times A'\to B\times B'\to A\times A'\to A$, and here $G\circ F\sim Id_{A\times A'}$, so the first half of desired statement is proven. Here $\iota_A$ is obviously includion.
Let's check that $f\circ g\sim Id_B$. Note that $f\circ g$ can be written as the composition $q\circ F\circ G\circ \iota_B:B\times f(a')\to B\times B'\to A\times A'\to B\times B'\to B$, and here $F\circ G\sim Id_{B\times B'}$, so the second half of desired statement is proven. Here $q$ is a projection of a product.
(sorry for this notation, but I cannot draw diagrams here)