This seems like a self-contradiction to me. If the wrong is in the service of rightness, if the ends truly do justify the means, then the "wrong" isn't wrong at all; it's right. And the "unethical" isn't unethical at all, it's ethical. The only reason people would think they're doing something "wrong" for the sake of rightness is because aren't very intelligent and fail to think fully contextually. They just think in terms of certain types of actions being categorically wrong or right. So the answer is, yes.
Being unjust in the quest for justice is a bit different, it seems. I think you can do something unjust for the sake of justness. If you kill an innocent person for the greater good, for example (and by the way, this is just an example--I'm not sure I'm actually a utilitarian), then it remains unjust for the killed individual. And it might break my heart to do an injustice for the sake of justice. I'm not even sure if that's possible, on second thought.. not because the just ends causes the means to be just, but because the unjust means causes the ends to be unjust. Not because utilitarianism is illegitimate, but because "justice" is the wrong word here. It's hard to think of an example where an injustice could serve a greater justice--that would be really reaching/an unlikely scenario--but it's easy to think of a case where an injustice could serve the greater good, utilitarianistically speaking. But something being in service to the greater good doesn't necessarily imply being an instance of justice, per se, imo. Justice is more about balance, or, as Rachel Dawes put it, harmony.