The big concern for tailwind/crosswind is inlet separation. Inlets are pretty much optimized for cruise operation - which means a relatively thin inlet lip. As a result they don't work all that great statically - the air from the side and behind the inlet is making a big, sudden turn over that inlet lip - the higher the power setting, the faster the air has to turn around the inlet lip. A tail wind or cross wind just makes that worse. If the airflow over that inlet lip separates, the resultant distortion into the front of the engine can result in an engine stall/surge. Some engines/inlets are more susceptible to this than others.
Rolls engines have another issue - both the RB211-524 series and the Trent series have issues with fan flutter when operating at low airspeed. In fact both the RB211-524G/H and Trent 800/1000 have incorporated 'keep out' zones that will prevent steady state operation in certain N1 speed ranges when the airspeed is less than 30 knots.
Starting is something else. Most modern (i.e. FADEC) engines don't have much of an issue, but some older engines (JT9D being a prime example) do. The problem is that when starting with a tail wind, it increases the back pressure on the core exhaust. Engines like the JT9D don't have much compressor stall margin during start to begin with - and engines typically lose some compressor stall margin as they deteriorate. The small amount of back pressure on the core due to a tail wind during start can push a marginal engine over the edge resulting in a 'start stall'.
Many years ago, one of the engine companies was using the very first 747 RA001 as a flying test bed for a new engine type. I was on board for engine ground runs prior to the first flight with the new engine type - the new engine worked just fine, but there was a tail wind and the old, tired JT9Ds simply refused to start until we got the aircraft turned around and pointed into the wind
