Wow. I never realized PoC was here. Fascinating. I would've come by a lot sooner if I had paid attention.
I have a somewhat unique perspective (well, unique if you're not a half Korean, half white army brat that grew up overseas for the majority of the first half of your life). I am a PoC, but I have my foot in a couple doors.
Race, in the military, has never been an issue. At least not race in the military overseas. We were always one big group of Americans all trying not to make Americans look bad and getting really pissed off at the ones that did.
I'm always incredibly embarrassed by white people and it never has anything to do with race. Sure, I suppose I'm embarrassed by racist white folks but I'm no more embarrassed by them than I am of the trashy disgusting white guy in a wife-beater drinking beer in his front yard on a beat up old couch and a toilet in the yard with old and dried out crispy flower carcasses because at one point they thought their busted toilet would make a great flower pot. As far as I'm concerned, they're all embarrassing and they all make us look bad.
I'm not black so my experience with racism is not quite the same - asians are slightly less offensive to racist whites than blacks, I think. Only slightly. I've had my share of having to duck while driving through Ludowici, Georgia. At the time, they were still burning crosses. I don't know if that's true today, though.
I've experienced ignorance from other cultures based on my race and my citizenship. And I've experienced racism from Koreans for not being pure. To many, I simply don't belong anywhere and even though I don't get acceptance (some Koreans can be really nasty about it) I do seem to at least deserve their pity - which I don't want or need. That kind of embarrasses me but I don't think anyone really sees it enough for me to be embarrassed for an entire race. I understand there's a lot of this in the black community for not being black enough.
I've also found that *all* people tend to forget I'm a PoC, too. Normal, don't consider themselves racist types. They get too comfortable around me and say things they wouldn't have otherwise said to a person more noticeably asian (or white when I'm with asians). In some ways, because of that I think I might actually be confronted with more racism than the average PoC.
Even with all that, I still don't actually get embarrassed by acts committed by race and I rarely look at things as being a possible representation of an entire race. Strangely, I
have wondered what other people like African Americans thought about it, or hispanics, etc. I've wondered how they respond to these types of things knowing how society views acts of violence/stupidity and applies it to the racial stereotype. I guess now I know.
With Asians I find there are less stereotypes about violence (well, if you forget about the Yakuza and the secret/stealthy/enlightened/deadly ninja/samurai/monk stereotypes), but more for social skills and appearance.
But I'm always embarrassed by acts representing my citizenship. So it's a different view but similar. I grew up feeling the pressure of foreign eyes surrounding me and judging me solely based on what country I was born in. And I saw the racism of those countries which isn't any different from the racism here.
I lived in Germany when the wall came down. I lived 45 minutes from the Czech border. You wouldn't believe the racism that came down with the walls, or the people that behaved in ways that only fed their stereotypes. And when the Gypsies came...hoo boy howdy...
So as someone else said, it's not just Americans that deal with it. We just don't really see it in other countries because...we aren't in other countries.
We were laughing to keep from crying. 
Instead it's somehow an overreaction on my part for being so silly as to be insulted by something that to me, was obviously geared to skew what I said. Pretending like it's not even a little understandable to you that I would take it the way I did, even if it was a misunderstanding, is beyond me. 
