The positive multiples of $ 63 $ are $ 63 $, $ 126 $, $ 189 $, .... All of these numbers except for $ 63 $ are bigger than $ 63 $. On the other hand, all the positive factors of $ 63 $ other than $ 63 $ are less than $ 63 $. So if Marie and Jay are thinking of the same positive number, that number must be $ 63 $. By the same reasoning, the list $ -63 $, $ -126 $, $ -189 $, ..., of negative multiples cannot include any factors of $ 63 $ except $ -63$. Thus the two possible values for the number Marie and Jay are thinking of are $ -63 $ and $ 63 $. The product of these two numbers is $ (-63)(63)=\boxed{-3969} $, since the product of a positive number and negative number is negative.