Name of current institution in a paper I'm about to publish a paper with my previous group. 

I'm not in a permanent position, and my boss doesn't care if I mention or if I don't mention my current institution in the paper.

For me, a researcher who wishes to have a permanent faculty position, would it be helpful or not at all to do that?
The work should be credited to the institute at which it was done - this is important for citation and publication metrics, and will be pulled out of the metadata of the publication by several services. The author is a separate entity to the institute which employed or supported the author. A subsequent change in affiliation - for contact details, e.g. - should be taken care of in a footnote in the front matter.