Why is the 95% confidence interval of the X intercept of linear regression asymmerical? Why doesn't linear regression report a standard error of the X intercept, like it does for the Y intercept?

The 95% confidence interval for the X-intercept is not symmetrical around the X-intercept. It goes further in one direction than the other, as illustrated in the graph below.

Follow the Y=0 baseline from left to right. The region between the 95% confidence bands for the best fit line (blue curves) is the 95% CI of the X intercept. You can see that this confidence interval (between the two outmost dotted lines) is not symmetrical around the X intercept (the middle dotted line).

This asymmetry will be very noticeable if you only have a few points with lots of scatter, and will be almost unnoticeable with lots of points with little scatter.

The linear regression analysis of GraphPad Prism reports the 95% confidence interval of the X intercept but not its SE. Because the uncertainty is not symmetrical, it rarely makes sense to report a standard error of the X-intercept. It is much better to report both ends of the 95% confidence interval, which Prism 4 shows you.

But if you really want to compute a single standard error for the X-intercept, you can do so by choosing nonlinear regression. Enter this user-defined equation:

Prism will report the best-fit value of the X intercept along with an SE (which won't be very meaningful).