Are there any footwear regulations for private pilots?
There is no policy on the books that I can find in regards to this. Loosely if flip-flops make it hard for you to operate the rudder it would fall under the catch all:


  § 91.13 Careless or reckless operation. 
  
  (a)Aircraft operations for the
  purpose of air navigation. No person may operate an aircraft in a
  careless or reckless manner so as to endanger the life or property of
  another.


However I was at an FAA WINGS lecture not all to long ago on preparing for accidents. My local FSDO's advice was to "dress for the crash" i.e. if you plan to fly over snowy terrain consider wearing boots etc. With this in mind flip-flops are more than likely not a great choice. The need to walk considerable distances over terrain is not unheard of after an accident.