This is actually a pretty advanced topic. There are a couple of possible shortcuts. If all Y variables are independent, you could simply run several chi-squared tests. You could also collapse your Y variables into a single new Y variable with 8 levels (y1-no; y2-no; y3-no
, y1-no; y2-no; y3-yes
, y1-no; y2-yes; y3-no
, etc.) and run a chi-squared test on the 2x8 contingency table. Neither of these are likely to be very satisfactory, however.
The preferred analysis is to construct a multi-way contingency table, and fit a log-linear model. The contingency table that you present is a two-way table with counts in rows and columns. With one X and three Y variables, you would have a four-way contingency table, where each cell holds the count of the number of times that combination of X, Y1, Y2, and Y3 values happened. Since it is hard to visualize a four-way table, it might also be convenient to display it as a flat table:
x-no; y1-no; y2-no; y3-no: n1111
x-no; y1-no; y2-no; y3-yes: n1112
x-no; y1-no; y2-yes; y3-no: n1121
x-no; y1-no; y2-yes; y3-yes: n1122
x-no; y1-yes; y2-no; y3-no: n1211
x-no; y1-yes; y2-no; y3-yes: n1212
x-no; y1-yes; y2-yes; y3-no: n1221
x-no; y1-yes; y2-yes; y3-yes: n1222
x-yes; y1-no; y2-no; y3-no: n2111
x-yes; y1-no; y2-no; y3-yes: n2112
x-yes; y1-no; y2-yes; y3-no: n2121
x-yes; y1-no; y2-yes; y3-yes: n2122
x-yes; y1-yes; y2-no; y3-no: n2211
x-yes; y1-yes; y2-no; y3-yes: n2212
x-yes; y1-yes; y2-yes; y3-no: n2221
x-yes; y1-yes; y2-yes; y3-yes: n2222
A log-linear model is essentially a Poisson regression model that compares models that take various combinations of the variables into account and compares their fit to the saturated model. If you use R, there is a nice tutorial here.