He said the threat did not come from a student at the school, and there was not much he could say while the police were investigating.
RNZ asked Killalea to confirm whether there was any association with white supremacists.
"That is the one thing that police have shared with us, that there could be an association with that," he said. "I can confirm the ... far right ... implication."
Some media have reported LGBTQI community members at the school were specifically targeted, but police said no specific group was targeted in the threat.
Killalea said a student had spoken to a staff member earlier in the week about a possible threat.
"There was nothing in that as far as we knew, there [were] no screenshots and no names, there was nothing actually to present to police.
"I want to be clear, we still don't have any link between that and the incident today, so we don't know whether they are related at all and the police don't know if they are related."
In an email to parents, Killalea wrote that "nothing that police reported to me this morning had any link with the Rainbow community and this should have been fact-checked with police".
"I do not know if there is any link and I am sure we will learn more in coming days - the only thing that police told me about the incident was that the person had posted images on social media that linked him to far-right and Nazi extremism."
Killalea told RNZ police warned the school about 8.30am of the threat and an officer was at the school during the lockdown.
He said the response from the students and staff was "fantastic".
"The response in the situation was absolutely what we would want. The kids did what they were asked to do with no fuss, followed all the instructions and acted appropriately.
"Everyone I came across was really calm and I went into a couple of classrooms and the kids were great - everyone responded really well."
Killalea said staff would be keeping an eye on students and taking care of them in the coming days in case some were shaken by what happened.
Police said anyone with information should contact them on 105, quoting file number 201126/3921.
- RNZ