Are there requirements for ice protection for general-aviation aircraft?
Does anyone know if regulation exists making it a requirement for general aviation aircraft to have ice protection systems?


In specific answer to your question: Yes, there are regulations for general aviation aircraft to have ice protection systems if the aircraft is to be flown
into known icing conditions.


FAR Part 23.2540 (applies to general aviation type aircraft) does specify certification criteria if the manufacturer is requesting approval
for flight into known 
icing conditions.
FAR Part 91, Subpart F- applying to Large and Turbine-powered mulitiengine airplanes and Fractional Ownership Program aircraft, does specifiy the 
conditions, equipment, etc. required to fly into known icing conditions. (business jets/turbo props, etc)
General Aviation aircraft operating under certain regulations, FAR 135.227 for example, are subject to specific regulations pertaining to operations into known icing conditions .
FAR Part 91.9, applying to all Part 91 operations, while not uniquely applicable to operations in icing conditions,
does require compliance with all operating limitations published in a flight manual, placard,
POH, etc.  So, if the POH for the aircraft in question prohibits flight into known icing conditions (as many general aviation aircraft manuals/POHs
do) then it cannot be done. However, there are also many general aviation aircraft that are permitted to fly into known icing conditions in accordance
with stipulations published in the POH (or similar).





  And any insights on why general aviation aircraft tend to use rubber boots and fluid based systems as opposed to electrothermal solutions?


The only insight I can offer regarding boots/fluid systems versus heated solutions is the manufacturer's choice based on performance considerations,
functional utility, and interoperablity with associated systems.  For example, an engine driven air pump for deice boots creates a performance overhead
that may not be worth the expense, assuming it's an option, based on how the airplane will be used.  Electrically heated props (heated near the hub)
are another option for many general aviation airplanes that can be added assuming certification for flight into icing exists.  Bottom line is, I don't
think there is a "preference" for general aviation airplanes, I believe design considerations and after-market options are the predominate factors underlying the type of deicing/anti-ice equipment available in most general aviation airplanes.  



Here is an excellent FAA Advisory Circular that pertains to your question:

AC 91-74B - Flight in Icing Conditions