I don't know about professional philosophers, but ignorant citations of random quotations rather than demonstrating a real understanding of the text makes people who actually have read the Republic (and thought deeply about it) really angry.
As the slashdot story itself wasn't about the Republic or Plato, I didn't feel it was appropriate for a multi-paragraph essay to provide an interpretation of the book just to correct someone who said something spectacularly wrong. But, since there still seems to be a lot of people who are claiming to know what's going on in the Republic, and they're really missing the market, I'll fill my comment out more fully. I'm only going to do this if I make one caveat clear: the Republic is probably the most widely debated book in the history of humanity. There's only 1 other that comes close, and that would be the Bible (new testament + old testament). And I'm willing to wager the Republic has been debated even more than the Bible, primarily because the Republic is heavily read and taken seriously by thinkers of all major western religions (Judaism, Christianity, and Islam), where the Bible may not be taken as seriously by certain sects of Judaism (new testament for example) or Islam. In short: No difference of interpretation about the Republic is going to be solved in a Slashdot comment.
Having said that, you would have to go through lots of interpretive hoops to claim that the Republic isn't about providing a definition for justice. And the surface (though not shallow) reading of the book is fairly explicit that Socrates reaches an exact definition of what justice is. And it is defined exactly as I said in my post: justice is minding one's own business and not being a busy body. The structure of the book is roughly as follows:
Book I of the Republic begins in traditional Socratic manner of Socrates running into various members of the Athenian community and beginning a discourse on some concept. In this case, it is about justice. Commonplace definitions of justice are provided by the various characters who make appearances. Cephalus defines justice as paying back your debts and telling the truth. Cephalus's son, Polemarchus, defines justice as helping your friends and hurting your enemies. Thrasymachus defines justice as whatever is to the advantage of the stronger, and then refines his position to say that justice is less profitable than injustice. Socrates finds all of these definitions to be inadequate and provides a rather shitty rejection of all of them. Thrasymachus gets pissed and says (in a rough paraphrase) "Socrates, stop screwing around and just tell us what you think justice is." Book I ends without any clear definition, and a rather unsatisfying result. This is traditional for the earlier Socratic dialogues, and it is sometimes believed that Book I was originally intended to be a stand alone piece titled the Thrasymachus.
After Book I, the style of the Republic changes dramatically. Instead of Socrates just rejecting positions, he actually begins to take a stance. This happens because Glaucon and Adiemantus (Plato's real life brothers) call Socrates out and say "You know, you haven't convinced me that Thrasymachus is wrong." Glaucon then suggests that the reason we concern ourselves with being just is that suffering injustice really sucks, and we'd all be better off if we agreed to do the just thing. However, Thrasymachus may be right, and doing what is unjust may lead to maximizing our own personal benefit in isolated cases. Glaucon then tells a rather famous story of a ring that turns one invisible, and the guy who gets it goes off and kills the king, seduces the queen and is living life pretty high. I don't know why turning invisible helps you seduce the queen, but whatever. That's what the book says.
At this point, Socrates suggests that they've been looking for justice in the individual, and since the individual is very small, it's hard to see it. So it would be best to blow it up to a situation that is much larger so you could see the details easier. Socrates assumes, without much argument, that justice in the individual soul is equivalent to justice in the city (which was the largest political unit of the day). Thus, Socrates says, if you can see the just city, it will help you understand the just soul. So Glaucon and Adiemantus start to help Socrates define what the best city would be.
From here on out, things get a little less straightforward, as they tend to jump around quite a bit. However, there are some major parts that happen in the rest of the book that help to explain what's going on. One is the explanation of the ideal city.
For Socrates (and Plato) the ideal city is one in which everyone has a specific task that they are good at, and the ideal city should have all of the four major greek virtues present. The four virtues are Temperance, Courage, Wisdom and Justice. So, he begins with the basics of the city. You need a class of people who are focused on production. These are farmers, craftsmen, traders, etc. These are people who care about making money and having a family. You need this part of the city if you are going to feed and clothe people. However, this is the lowest form of human activity, so they don't really represent any distinct virtue. But, you can instill within the working classes a sense of moderation, which can give them the virtue of Temperance.
The second class that the city needs in order to be a perfect city is a military class. The primary purpose of the military class seems to be to attack other city to steal with luxury items. (I'm not kidding, the military gets introduced because Glaucon is upset that the ideal city of just workers is lacking in relishes). Socrates gives into this for some reason and suggests that it would be a good idea to have a military class that governs over the city to moderate the producers (providing Temperance), and since the military needs to be able to successfully fight battles, they will embody the virtue of Courage. We now have two of the major virtues in our ideal/perfect city.
At this point, Socrates effectively tells a joke and says something to the effect of "We should let philosophers be kings of this city." The idea being that philosophers represent wisdom. (Philosopher being derived from the greek words for 'love' and 'wisdom' indicating that they are lovers of wisdom). With philosophers as kings, you will have the wise ruling over the courageous and the moderate, accomplishing three of the four major virtues in the ideal city.
At this point, Socrates says (again paraphrasing) "Justice has been under our nose the whole time! Justice is accomplished when everyone does what they are best suited to do, and nothing more than that." Or, and as I quoted before, another way of putting this is "justice is not being a busybody." When business men try and run cities, they seriously mess things up, and that's unjust. Likewise, when the military class is trying to maximize their own personal profit, they become seriously corrupt and mess things up. That too is unjust. So the definition that Plato provides, _explicitly_ in the Republic is justice is when one minds their own business and is not a busybody.
He then suggests this can shrink down to the individual soul level too. You are a just individual when your appetites are moderated (you do not want too much, or too little), when you are courageous and capable of facing adversity (without being foolhardy or cowardly), and when you govern yourself according to your reason (which involves being wise). If you have all of these virtues in your soul, and they are doing what they are supposed to be doing, then you are just.
There are plenty of good arguments to say that Plato is being metaphoric with some of this stuff. However, as I asserted before (and directly quoted from the text), Plato explicitly defines justice as not being a busybody. He does this in his most important political work that begins (and runs throughout) on the subject of justice.