Does selfishness increase in periods of extreme anxiety Does an individual going through a period of extreme anxiety, show increased levels of selfishness during that period ? 

When this period ends, does the individual returns back to normal levels of selfishness ?

The following extract is from this  article, which describes the personal experience of a person with anxiety/depression.


  When we’re anxious, we become lost in our own thoughts, waging a
  battle so distracting in its intensity that it makes it nearly
  impossible to notice the outside world. It’s not that we don’t care,
  but rather that we’re so distracted by our discomfort that it is only
  with great effort that we’re able to look out and truly see others.


This somehow verifies my intuition, that extreme anxiety increases selfshiness. There is so much anxiety related activity in the head, that other aspects like other people, are neglected.

This is just a personal report though, I need more solid research data.
It is hard to argue that anxiety makes you more "selfish". Your definition of "extreme periods anxiety" is rather vague and hard to operationalize, same goes for selfishness in this context. The article you cited is not based on scientific literature but much rather a personal look on anxiety and selfishness. Therefore it is in no way conclusive evidence. 

Anxiety is a natural body response to certain dangers. When anxiety becomes abnormal it can  be diagnosed as an Anxiety disorder (see DSM V for criteria for certain anxiety disorders). There has been a lot of research on General Anxiety Disorder (one of the anxiety disorders) and the worry-process. However, too my knowledge there are hardly any studies that describe individuals become more selfish. Like I said, this is also definition question. Certainly people with anxiety worry more and are more focused on the thoughts in their head. However, as far as I know there is no convincing evidence that people with strong anxiety do not care about others except for themselves. On the contrary some people with Anxiety disorders excessively call their loved ones to make sure that they are alright.

Further reading: Butcher, Hooley and Mineka, Abnormal Psychology 16th Edition, Chapter 6.