A judge only trial (aka a bench trial) is possible in the United States, though it must be requested by the defense (essentially the defense waives its right to trial by jury, in which case the judge acts as both trier of law (his or her usual role) and trier of fact (the jury's usual role).
A court case may be partially sealed so as to discuss evidence of a sensitive nature, such as classified evidence or testimony by a witness whose identity needs to be concealed (i.e. a child or still undercover cop).
I am not finding any source beyond Snowden's statements (either directly or through his legal council) stating that Snowden will have a secret mandatory bench trial, nor can I find anything specific to the Espionage Act that mandates these conditions (in fact, the U.S. has elected to not prosecute some accused violaters because they would have to show classified information in open court). The lone mention of a bench trial was related to the trial of PFC Manning, whose court martial was a bench only trial, but this was due to request by Manning and not a matter of the law itself. Manning's trial did have moments where spectators were asked to leave as classified evidence was going to be discussed, but the trial was followed by the media very closely at the time.
The Manning trial could be where Snowden recieved the idea that he would be in a sealed court bench trial, but it is also likely that he is using it as deception to cover his reason from seeking asylum rather than face charges.
The only other spot of confusion in your link is discussion of the charges being sealed. This happens frequently in the U.S. Federal Court and are used to delay the arrest of the accused for various reasons. I'm unfamiliar with some typical reasons why, but it's not that uncommon for federal prosecutors to unseal charges after they have been approved by a Grand Jury.
EDIT: Following your updates I did some digging:
First, the Nation article links to an NPR article, which sources a book written by Snowden, so the original source is still Snowden. Second, as discussed in the NPR article cited, Snowden is not being denied a trial by jury but rather the jury will not be allowed to hear his defense.
This distinction requires some explanation: In the United States (and common law), intent is not always an element of a crime (some crimes you need to show that the action that was taken was done so with a mindset by the actor such that he intended the actions to be criminal in nature). Usually intent matters in cases of corruption or obstruction of justice (just because you don't tell the cops something important or destroy evidence before they can process it does not mean it's obstruction. You could just be stupid or just not helpful because it's your right to shut up when the cops ask you something. It's only if the un-helpfulness was done in a way where you knew it would conceal a crime being investigated). Violations of the Espionage Act are not an intent based crime, which means you are answerable to it even if you were trying to do something good with it.
Basically, per the NPR article, Snowden wants the jury to consider the reason why he did what he is accused of... which the law doesn't allow him to do. When the jury is given the case, they will be asked to find guilt on the question of "Did he do it?" (which Snowden said he did numerous times) and not "Did he mean well when he did it?" (debatable, but not legally required to be answered to find guilt.).
Intent usually comes into play only in the sentencing phase of the trial, but this phase only starts once the jury says someone is guilty of something. It's here where Snowden will be allowed to testify to the reason why he did what he did, but since he has to be found guilty, this doesn't keep him out of jail... it only lets a judge who is sympathetic give a lesser sentence.
Snowden will be allowed a jury, but he isn't allowed to tell the jury why he thinks it's right or wrong. This is important because the jury does have the power of jury nullification, which is basically allowing the jury to find that the defendant is innocent not because he didn't do it, but because there shouldn't have been a law that criminalizes the action in the first place (it's kind of confusing, but it can possibly overturn the law). However, there is a catch in that the jury must decide this on their own, and neither party may raise the idea up to the jury during a trial as per the rules of the courtroom procedure. Snowden is arguing that he wants to invoke jury nullification in his defense (without saying it, but that is what his argument is) but since he's not allowed to do it, he's claiming he is not being allowed to present his side to the jury.
The whole intent thing is reason behind the famous "I was just following orders" line from the Numberg trials. In German civil law, intent in a crime does come into play in the trial phase, not the sentencing, and in German Law a murder committed without personal motives against the victim is a lesser crime than murder with personal motives. The German soldiers were arguing that they didn't commit the more serious crime accused because they were impersonal with their victims with the "just following orders line". Unfortunately, the trials were being conducted by two countries who used Common Law (U.S., and U.K., which ask "did you do it?" when finding guilt and save "Why?" for after they found it) and the other two countries (France and the Soviet Union) were all too happy to use this system as the French were pissed over the occupation and the Soviets were pissed that anyone was bothering with a trial and not just hanging the Nazis right now (I kid here, but the Germans by and large would rather the U.S. and U.K. handle their surrenders because of their distinct lacking of Stalin any where remotely close to their criminal justice system.).