Are there any restrictions on long overwater flights by private aircraft? It is talked about in many questions here that (with the exception of older 3+ engine aircraft) airlines and their planes must be ETOPS certified if they are going to fly more than 60 minutes away from a suitable diversion airport.

Are there any similar restrictions for private flights?

If regulations differ, assume the flight is between the US and Europe in a US or European registered plane.
I am unable to find any ETOPS restrictions for private aircraft in the FAA FAR/AIM Part 91. The FAA Advisory Circular AC 91-70B is applicable for commercial aircraft operators. Anecdotally, general Aviation single-engine aircraft (Including helicopters) make oceanic journeys on a regular basis.