I don't see why you would not, so long as it is in the proper context and not as a substitute for actual qualifications and high quality research outputs and teaching experience.
Almost all posts in academic institutions now come with the expectation that you work will have "impact" outside of academia. For people like me who work in astrophysics, and I guess for many who work in areas like particle physics or theory, then it is rather difficult to point to major technological breakthroughs or start-up companies who are going to take your work and produce a new best-thing-since-sliced-bread device. In fact, if your research is more "blue skies", what you have to argue is that you are reaching out to wide and diverse audiences and having an impact in that way.
Unfortunately, stack exchange does not really fulfil that objective because mostly you are not writing about your own research (I would guess only 1-2% of my answers on Physics or Astronomy SE are linked to my research) and the audience is not that diverse and difficult to quantify. Creating a personal blog about your research area might be more effective in that respect, though I doubt I would have gotten 3 million views of a blog about my research in 6 years.
What stack exchange does do however (providing that you have written many highly upvoted answers), is demonstrate the ability to communicate about science/mathematics/computing to a variety of audiences and that may be a very important part of the role you are applying for. It may also demonstrate a passion for teaching and often demonstrates the ability to marshall facts and research from a variety of sources to support an argument. So you could certainly use stack exchange as an example of work that you have done which might illustrate your ability or potential for certain kinds of tasks.
Having said all that, if you are answering many questions per day on average, then it probably just shows you're spending too much time on stack exchange... like most things, moderation is probably to be advised.
Bottom line: I don't do SE because I think it adds anything particularly significant to my CV nor am I especially altruistic; I do it because I find it entertaining; it drives me to look at old problems in a different way; it often gives me interesting ideas for student problems and assessments; and has prompted me on many occasions to keep up with the latest developments in some areas. I do add it to my professional CV, in the section about all the other public outreach work I do and I regard it as part of that.