All three of these Supreme Court cases have to do with freedom of the press. Near v. Minnesota was about whether the state government could restrict the publication of a "malicious, scandalous, or defamatory" newspaper by those previously convicted of libel. New York Times Co. v. United States was about whether the US government could restrict the Times from publishing the Pentagon Papers, a series of leaked documents detailing US war crimes in Vietnam. Hazelwood School District v. Kuhlmeier was about whether or not schools could restrict student speech in the context of school newspapers.
All of these cases, specifically, have to do with the concept of "prior restraint." The US Constitution says that congress shall make no law abridging the freedom of speech or of the press. That means just what it says: you can't throw someone in jail for publishing something you don't like. The tricky part is where speech is itself action. To use the most famous example, "yelling fire in a crowded theater" isn't a matter of free speech: it is a matter of not creating a panic by spreading disinformation. If you do yell "Fire!" in a crowded building and there is a panic as everyone rushes to get out and someone is trampled, you are, in fact, responsible. So, this raises an important question: can the government stop you ahead of time? To extend the example, if a policeman knew that you were going to go into a theater to yell "Fire!" as a prank and stopped you from going in, would that be allowed? Or would the policeman have to simply wait until you were done and then arrest you for having caused a panic? That's the question of prior restraint: if the government can foresee negative effects from your speech, effects that it would be otherwise lawful to punish, can the government prevent you from making that speech in the first place? This is especially a question in the first two cases, where there is a clear criminal act: libel (deliberately spreading misinformation) in the case of Near v. Minnesota and leaking classified documents in the case of New York Times Co. v. United States.
In two out of three of these cases (Near v. Minnesota and New York Times Co. v. United States), the Court ruled against prior restraint. Even though libel and leaking classified documents are illegal, The Saturday Press and the New York Times couldn't be stopped, ahead of time, from publishing whatever they wanted. They could only be punished later if what they said turned out to be illegal. In the case of Hazelwood School District v. Kuhlmeier, the Court avoided overturning this precedent by saying that, contrary to the students, the case wasn't about prior restraint at all but rather about whether a school newspaper really was a public forum (they ruled that it wasn't and that that was why schools could exercise censorship in this case). So the common thread between these three cases is no prior restraint.
Now, what does all this have to do with the Bible? It is unclear what, if any, biblical justification there is for the First Amendment. The first and second books of Kings, for example, consistently use whether a monarch restrains the worship of idols in the land or not to be a measure of whether he is a good or a bad king. For example,
Asa did what was right in the sight of the Lord, as his father David had done. He put away the male temple prostitutes out of the land, and removed all the idols that his ancestors had made.
(1 Kings 15:11–12)
So, the idea of a government that doesn't exercise any prior restraint on people's expression of ideas, including religious ideas, is not really a biblical concept. The question we have to ask, therefore, is whether or not the norms laid out in the Bible for Israel's kings should apply to the rulers of other nations? How much is the Bible describing things that the kings of Israel are, specifically, supposed to do because they are ruling a chosen people, and how much is it simply laying out what good rulers are supposed to do in general? If the answer is the former, then there may be some biblical justification for free speech. If the answer is the latter, then the Court really cannot be said to have upheld "biblical" standards.