OK, I just wasted an hour and now need to go to bed.
17 players have beaten all of the Big Four. In order of total wins:
20 Del Potro
19 Wawrinka, Berdych
18 Tsonga
14 Roddick, Davydenko
13 Nalbandian
10 Ljubicic
9 Nishikori
8 Gonzalez, Thiem, Cilic
7 Clement
6 Soderling, Ancic, Simon, Kyrgios
But here's something interesting: of those 17 players, Kyrgios has the highest win percentage (albeit in a relatively small number of matches):
35.3% (6-11) Kyrgios
34.2% (13-25) Nalbandian
32.0% (8-17) Thiem
31.8% (7-15) Clement, Ancic
28.2% (20-51) Del Potro
28.0% (14-36) Davydenko
27.6% (8-21) Gonzalez
26.5% (18-50) Tsonga
25.9% (14-40) Roddick
24.4% (10-31) Ljubicic
21.8% (19-68) Wawrinka
20.2% (19-75) Berdych
18.0% (9-41) Nishikori
16.2% (6-31) Soderling
15.4% (8-44) Cilic
12.5% (6-42) Simon
A selection of players who played all four but beat only three of them:
Didn't beat Federer: Ferrer, Verdasco, Dimitrov
Didn't beat Nadal: Safin, Gasquet, Anderson
Didn't beat Djokovic: Monfils, Raonic, Coric
Didn't beat Murray: Hewitt, Ferrero, Goffin
I'm sure there are more, but those are a sampling that I found. Coric is the youngest - he just needs a win over Novak ("just"), and then becomes the youngest to beat all four. Sascha Zverev has only beaten Roger and Novak and probably won't get a chance to player Andy again.
Poor Berdych: played the Big Four a whopping (record?) 94 times. He might go down as the only player in history to play all four a total of 100+ times.