When is my biennial flight review due? I logged into my flying club's online scheduling website today (May 19, 2015) and it was warning me that my Flight Review was about to expire (on May 31, 2015). My last flight review was on June 7, 2013.

According to 14 CFR 61.56 (c), the look-back period runs from (my emphasis added):


  ... the beginning of the 24th calendar month before the month in
  which that pilot acts as pilot in command ...


So am I right in calculating that my last Flight Review is actually valid through June 30, 2015?
Yes. Your interpretation is the correct one. Several FAA legal interpretations cover the "calendar months" phrasing; the best I could find was the Harvey interpretation from 2012:


  Based on the facts that you have presented, a biennial flight review accomplished on April 2, 2002, would not lapse until May 1, 2004.


Basically, your flight review is legal until the end of the month (ie, June, or August) it was completed in two years before. The same calendar logic applies to maintenance like the annual inspection.