It's an interesting constitutional case, and the First Amendment will undoubtedly be raised, because there are so many precedents involving it, but not on its merits. The First Amendment only protects the right to express and to communicate, not to withhold information. There is a right to privacy in the Ninth Amendment, but it is qualified by the public's right to know, on a question by question basis. It is like discovery in a civil case. The witness, whether before a grand jury or in a deposition, has the right to object to particular questions, and the matter may then be referred to the judge for a decision on whether the public's right to know prevails over the witnesses' right to privacy. That decision may, of course, be appealed to a higher court.
The only right to withhold information is in the Fifth Amendment, but, contrary to established precedents, that originally protected against not only being required to answer a question in a criminal case, or in the situation in which the witness might be incriminated thereby, but also questions that if answered truthfully, might tend to impair the witnesses' reputation or property interests. And, contrary to established precedents, the right originally applied on a question-by-question basis, not to testimony generally. In other words, a witness does not waive his right to withhold answers to some questions just because he agrees to testify on others.
See Shelton v. American Motors Corp., 805 F.2d 1323,1327 (8th Cir. 1986), which held that a party should not be allowed to depose opposing counsel without demonstrating that (1) no other means exists to obtain the information; (2) the information sought is relevant and not privileged; and (3) the information is crucial to the party’s case.
In Hart v. Frost, 73 Okl. 148, 175 P. 257 (1918) the judge held:
The court finds that in the case at bar the court prevented defense counsel from objecting to each question as it was asked . . . by overruling the defense's request to do so, even though the transcript does not specifically show what objections were overruled, interposed by the defendant. The court specifically recalls the nature of the statements made in chambers off the record . . . and it is the court's finding that all parties understood that the defendant's objection that was overruled was his request to interpose objections question by question . . . .
Also see United States v. Allee, 888 F.2d 208, 212 (1st Cir. 1989).
It is important to understand that the privilege of confidentiality is just that, a privilege. It has no basis in the Constitution. It is a matter of customary prudential deference, and offers the argument made above that it should be respected if there are other ways to obtain the same information, and only breached after it has been demonstrated that no other alternative is available.
This applies to any confidentiality custom, whether it be for a journalist, a religious counselor, a physician, or whatever. It is a matter of judicial discretion, and is based only on custom, the resistance that some kinds of witnesses can be expected to offer, and the support of it by powerful political groups.
Of more interest to me, as a constitutional historian, is the way the grand jury often is abused by being used as an inquisitorial tool by prosecutors. If a witness finds that the prosecutor is present in sessions of the grand jury, he should demand his right to be questioned by an independent grand jury which is not unduly influenced by a prosecutor or the judge, and to demand they not be present. Only the members of the grand jury and the witness should be in the room during questioning.
A weaker but potentially useful line of attack, as a political weapon if nothing else, is to challenge the statute which the grand jury is apparently attempting to enforce, namely, the statutory basis for charging someone with a crime for disclosing the identity of a federal agent. There is no constitutional authority for such a statute to be enforced against civilians, only against government personnel who have agreed to be subject to it, such as military personnel and militia in actual federal service.
The Treason Clause might be invoked for authority, but the answer to that is that there is no enemy declared by resolution of Congress, as required by the Constitution. No declared enemy, no applicability of providing "aid and comfort" to an enemy.
One may object that this doctrine of interpretation would leave us defenseless to real enemies. The answer is that the Constitution arguably needs to be amended to provide that power, but unless or until it is amended, no such authority exists.
Usual disclaimer: This is not legal advice. Using these arguments is unlikely to result in favorable outcomes. However, as a matter of original understanding and true law, they are correct. Just don't expect appeals to the Constitution as originally understood to be respected by today's courts.