Can a supersonic aircraft have winglets?
This has been (and thus can be) done and there is some current research on it as well. The XB-70 Valkyrie had dropping wing tips that added stability at high Mach speeds as well as allowed the plane to ride on its own shock wave. It also effected trim drag, 


  The repositioned wingtips also reduced the area behind the airplane's
  center of gravity, which reduced trim drag. The downturned outer
  panels also provided more vertical surface to improve directional
  stability at high Mach numbers.


You can check out this podcast episode that interviews some of the people at NASA doing various research projects including wingtip control systems and other interesting applications in that space. 

You may find some info in this book on supersonic wing theory. 

This thread (and you know how accurate the internet is) has some interesting points on the matter.