Questionable Content 3923 has a caption that reminded me of your question:
Either they don't have blush panels or they're very good at controlling it
I don't know how canonical these captions are, but it seems to be a word of god implication that some robots and AnthroPCs have specific hardware that allows them to blush.
As for why, I think there's an assumption in the question:
What possible advantage could it be to this machine to be able to blush?
The assumption is that a robot would only have the ability to blush if it gave them some advantage. I don't think that's necessarily the case.
But the short answer is: the ability to blush gives robots the same advantage it gives humans.
Keep in mind that robots in Questionable Content are sentient, and have full lives. They aren't just mindless machines designed for one purpose. They also have some control over their appearance: we've seen several AIs choose bodies and body parts for themselves. So it could be that nobody gave Bubbles the ability to blush. She might have chosen that ability for herself.
On top of that, I think Adamant made some good points in the comments: You can think of blushing as a form of nonverbal communication. According to Wikipedia, nonverbal communication represents two-thirds of all communications. So having the ability to communicate nonverbally is useful in general. Blushing can communicate embarrassment, or anger, or sadness. It could even be a part of a more intricate system that includes other similar abilities.
You've asked why a military robot would need the ability to blush. I think the answer mostly lies in the first part of this answer (they're sentient and not just mindless killing machines), but from a purely utilitarian perspective, keep in mind that being in the military doesn't just mean you're constantly out killing enemy soldiers. It involves tons of communication: with your squadmates, with your reports, with your commanders, with civilians, and even with enemies.
You might ask similar questions: Why can Bubbles develop romantic feelings? Why can Bubbles feel self-conscious? Why does she have hair? Why can she laugh? The answer is because she's not just a mindless killing machine. She's a full individual.
Edit: reffu pointed out that this is mentioned in Questionable Content 3632:
Tilly: I apologize if this is inappropriate, but why are robots able to blush?
Winslow: They say it helps humans empathize with us, but I think the designers just think it's cute.
Also note the caption of that comic:
veering dangerously close to the 4th wall there, Jacques
This might provide some insight into the out-of-universe answer: they can blush because Jeph Jacques thinks it's cute.
Another edit: Questionable Content 4577 contains some support for the above theories:
Clinton: And like, some AIs prioritize utility, like if you're a construction robot you might wanna be big and have a backhoe arm, whereas other AIs place more emphasis on aesthetics, like if you wanna be human standard or a weird spider or something because you think it looks cool.
So, again, Bubbles can blush because she chose a body that can blush.