This is a fairly straightforward achievement, yet there seems to be some confusion surrounding whether or not drop-in games count. They don't. You need to complete 30 EASHL Club Matches for the achievement to pop. It popped for me upon completion of my 30th club game (40th overall game), right when i exited the game and returned to the position select screen.
As confirmed in the comments below by Potent Skittles, these games DO NOT have to be consecutive. (the opponent quitting out won't hurt your progress). All that matters is you complete 30 club games without quitting. The easiest way to ensure you get this achievement is to simply never quit out of club games - take the loss like a man and move on.