The wording of the oath is seemingly clear: a witness swears to "tell the truth," and thus not lie; "the whole truth," and thus not omit pertinent information; and "nothing but the truth," and thus not obscure their true statement. If Mueller is to comply with the oath he will have to swear, he would seemingly have to provide all relevant information.
However! The real issue is not the wording of the oath, but whether Mueller can legally be compelled not to omit any information. In other words, could he be found guilty of perjury under federal law if he didn't tell the whole truth?
The precedent of Bronston v. United States (1973) suggests that he could not, and that omitting information (as opposed to outright lying) is not illegal. In this case, a witness omitted relevant information, and in fact directly answered a question with information that did not answer the question.The judge ruled that this was not perjury.
Per Wikipedia, quoting the ruling, a witness is not susceptible to prosecution for evasion; rather, it is the lawyer's job to prevent this:
"If a witness evades, it is the lawyer's responsibility to recognize
the evasion and to bring the witness back to the mark, to flush out
the whole truth with the tools of adversary examination."
And the motives for evasion are irrelevant:
"To hold otherwise would be to inject a new and confusing element into
the adversary testimonial system we know. Witnesses would be unsure of
the extent of their responsibility for the misunderstandings and
inadequacies of examiners, and might well fear having that
responsibility tested by a jury under the vague rubric of "intent to
mislead" or "perjury by implication."
The article also notes that the case has not been overturned or rendered inapplicable:
The decision has been cited in many cases since then and has become the controlling legal standard of perjury in federal jurisprudence. It was invoked during Bill Clinton's impeachment proceedings in 1998 as a defense to charges of perjury against him.
It has long been criticized for the loophole it creates in the perjury
statutes as essentially allowing a witness to lie without
consequences. Nevertheless, later Courts have refused to overrule or
otherwise limit it despite some moves in that direction by lower
As such, Mueller could omit whatever information he chose, and, as long as he did not lie outright, would not be subject to the threat of prosecution.
If he outright refused to answer a question, however, it is possible that he could be found in contempt of Congress and subjected to relatively light penalties. If his questioners are careful, it's possible that he could be put in a situation where he must either answer truthfully, lie (and thus commit perjury) or outright not answer and possibly held in contempt. However, if Congress doesn't know what to ask, Mueller could certainly legally omit relevant information.
Further, there's also the political question of how likely Congress actually would be to pursue a contempt charge or perjury case against Mueller, in the off chance that they could. Mueller is popular among Democrats, and Trump would surely welcome any charges against him, which would probably convince many voters that Trump had done nothing wrong—after all, if even the Democrats think Mueller's up to no good, people would think, there must be something to it—and bolster his approval ratings, which were damaged by the release of the report. I can't see this being a route the Democratic-controlled House would like to pursue, even if Mueller outright lied.