Overwatch - Winston Vs. Doomfist in a Nutshell

Technically not correct. Reaper crushing the glasses pissed off Winston because they used to belong to Dr. Winston, his teacher and father-figure. He takes them off in the Doomfist cinematic so that he doesn't break them by accident when he unleashes his anger in the following fight.

No, the comic only implies that Doomfist broke the glasses because it's a reference to One Punch Man. In that scene, Saitama breaks Boros's armor, and the equivalent to that would be Winston's glasses.
Now that I've explained the joke, I'm off to go arrange its funeral.