Best current player in the endgame? In one interview, I do not remember when (please if someone knows give the link), Kramnik quoted that the best endgame player is Alexei Shirov... I think this interview was around year 2000. Obviously for anyone to reach the very top level they must be very strong in the endgame. Now, I do not think anyone would support Shirov as the best in endgame because he is not top 20 anymore and the world champion extracts most of his wins in the endgame.

Who is the strongest player in endgame nowadays?
The blatantly obvious answer would be Magnus Carlsen who is also the No.1 player in the world, his ability to squeeze out wins from drawn endings is unsurpassed (with a few exceptions, e.g Andersson and Smyslov). Other than that I would naturally consider Vladimir Kramnik, although it is very hard to draw the lines between the various stages of the game nowadays as most of the top players nowadays are extremely versatile and have a universal style of play.