Have most journals recently stopped providing off-prints to authors?
Off-prints are a remnant from the days when photocopying hadn't been invented and, if you wanted your own copy of a paper, the only reasonable way to get one was to write to the author and ask for an off-print.

Providing off-prints to authors certainly seems to be becoming less common.  Some journals still provide them for free, some only for a fee, and some not at all.  I don't think most people care, and among those that do care, many prefer not to receive the off-prints.  It's been years since I received a request for an off-print, so when I do get them they just end up sitting in piles in my office while I offer them as party favors to anyone who enters the office.  Some decline and probably many of the rest recycle them, since electronic copies are far more convenient.


  Can I pressure the journal into sending me off-prints free of charge (e.g would trying to refuse signing the publishing agreement, unless they provide the off-prints for free work?).


I wouldn't try pressuring them, which could come across as both eccentric and rude.  Instead, you could try begging, by explaining that you are a postdoc with strictly limited funds but would really love off-prints and hope they could provide them at a reduced cost.  I have no idea whether this could work, but the worst that can happen is that they'll say no.


  Have people tried boycotting journals not offering off-prints?


You are welcome to investigate which journals provide off-prints for free and submit your papers there, but I doubt many people will join you in this.