Should self-citations be excluded when calculating the h-index?
First and foremost, I recommend reading the related question: "Why is it said that judging a paper by citation count is a bad idea?"  That question may help relieve some of your concern about the importance of having an answer for this question.

Now, turning to H-index: to the best of my knowledge, there is no consensus as to whether self-citations should count or not count.  Saying whether self-citations are included might be useful, but then it would also be useful to know quite a bit more about how the database for computing an H-index is being constructed.

My own thinking on pros and cons goes as follows:


Omitting self-citations means you get a clearer picture of whether other people are paying attention to one's work.
Self-citation is also entirely appropriate and legitimate and at moderate levels can be a good indicator of a healthy and ongoing research program.
Defining self-citation is not entirely obvious, when co-authors are taken into account.  Consider the following: if A and B co-author a paper, and then B cites the paper, should that count as a self-citation for A too?  It's not A who is citing, but the citation still might be "discounted."
Finally, precise values of H-index are not very valuable in any case, since bibliometrics are not very good at evaluating scientific impact.


Given all of these things, I personally think the best thing to do is to count self-citation in H-index and mark it clearly as such.