Can I include online reading guidance as research experience? I graduated back in 2015 and have been serving my military obligation. Though I was off from academia, time to time, I kept in contact with former research supervisors. They gave me a list of papers to read and when I am free in the army, I read some and reproduced results of the papers in the list. I even ended up publishing a paper with my supervisors.

As my service ends in early 2018, I plan to apply for Ph.D. programs this fall and I am wondering if the activity I described above can be seen as a research experience. If so, I would like to put it on my CV, however, unlike my other experience, which was done in a specific institution as a research intern, I had no affiliation nor was a member of any institution. How can I elaborate my activity in CV?
This is something I would just explain like you did in this question. From a potential supervisor's perspective, it is important to know that a PhD candidate is willing to go an extra mile and you clearly did that. Since a paper came out of this, you even have some evidence that you did not make up a nice story to impress someone, but you did actual research work. 

I would just put a line in the cv like "independent research work jointly with xxxx, resulting in publication [yyy]".