Unfortunately, the law on this is a tad vague.
There have been two important cases which have distinguished and narrowed
Pickering:
Connick v. Myers and
Garcetti v. Ceballos.
Connick v. Myers held that even when an employee is speaking on a matter of public concern, discipline or termination may still be justified if the government's interest in the effective and efficient fulfillment of its responsibilities to the public outweighs the employee's free speech interest. In that case, the complained-of speech constituted at least some degree of insubordination, and the Court held that the government-employer was entitled to some degree of deference in maintaining working relationships in the office. (As a side note, the Connick in that case is New Orleans DA Harry Connick, musicial Harry Connick Jr.'s father).
Garcetti v. Ceballos, decided in 2008, held that public employees speaking in their official capacity have
no personal First Amendment protection from termination for that speech.
Of course, those are both free speech clause cases, and I think you raised the possibility of a Free Exercise Clause claim. On that, the most recent case I'm aware of is
Employment Division v. Smith, where the Supreme Court held that it was constitutionally permissible for Oregon to fire an employee for using Peyote off the job, even though the employee was a Native American who was consuming Peyote for religious purposes.
Trying to come up with a clear rule from all of these cases is difficult, but I think, in this case, it comes down to one question: Does the employee's off-duty public Koran burning impede the NJ Transit Authority's ability to effectively and efficiently fulfill its public responsibilities?
Eugene Volokh doesn't think so, but I am not sure I agree. Unlike
Pickering, the speech here was highly inflammatory (no pun intended) and directed at a religion adhered to by a significant number of New Jerseans who are served by the Transit Authority, and public book-burning is generally considered unruly and distasteful behavior. I think the NJTA (or whatever their initialism is) has a pretty strong argument that they were within their discretion. It could certainly be said to affect his ability to function in the work place as much as getting high on Peyote over the weekend does.
On a personal note, as a public employee myself, I would certainly expect to be fired if I participated in a public Koran-burning. Of course, I am a judicial employee and my immediate boss is an elected official.
Also, I do have one question that bears consideration: If Mr. Frenton had participated in Fred Phelps' other favorite activity, protesting at servicemen's funerals, would anyone here be questioning the NJ Transit Authority's right to fire him for
that? I wouldn't. Protesting at a funeral, like burning the holy book of a major religion, is just plain unseemly behavior. I don't think it is unreasonable for a public employee to be expected to exercise his right to free speech with at least a minimum of public decorum. I don't think the courts will, either.
"If Mr. Frenton had participated in Fred Phelps' other favorite activity, protesting at servicemen's funerals, would anyone here be questioning the NJ Transit Authority's right to fire him for that? I wouldn't. You mean anyone other than the ACLU? Probably not, but I'd wager a week's pay the ACLU would. I wonder if they'll assert themselves here.
In my mind, I keep coming back to this. If freedom of religion gives freedom to praise our supernatural deities, must it also protect those who wish to blaspheme those same supernatural deities, does it not?
Like I said, my experience so so painfully abbreviated in this are of law, I recognize that what might seem logical (not firing public employees for exercising their 1A rights), may very well be plainly legal (or illegal).
"Eugene Volokh doesn't think so, but I am not sure I agree." Thanks for the link. I finally had some time to read all the comments. For that site, the comments are remarkably thin with respect to case law citations, affirming what you asserted earlier - the law is a tad vague.
There is one thing that I'm surely confident about - this issue isn't going away anytime soon. I'm not sure this is the case that will settle the law, but the writing is on the wall. There will be more of these kinds of retaliatory events, and that will sure prod the judiciary to act.