On Monday, the Welsh Ambulance Service described the situation as "very rare".
Speaking to Radio Wales Breakfast, Mr Killens explained that declaring a critical incident "signals the the rest of the NHS that we've got a problem, we are under extreme pressure and we need help".
Health boards in Wales have reiterated the ambulance service's message to only call 999 or come to the hospital with life-threatening symptoms or in a emergency.
A spokesperson for Cwm Taf Morgannwg University Health Board said their hospitals were extremely busy.
They said: "None of us wants to start 2025 in a busy emergency department or in the back of an ambulance so please take care as you celebrate the new year and think carefully about whether your health need is an emergency."
A spokesperson for Hywel Dda University Health Board said their hospital sites, especially emergency departments, were operating under "extreme pressure".
A spokesperson for Cardiff and Vale University Health Board said it was experiencing operational pressures following the Christmas break.
"The increase in flu both in the community and across our hospital sites has resulted in several ward closures with 100 beds currently closed of which 10 are empty - we expect this to increase over the next few days," they said.
A spokesman for Aneurin Bevan University Health Board said their services were facing the same extreme pressures that was being reported across Wales and the UK.
"We are currently operating at a level 4 escalation status, with the demands prevalent across our entire system and unfortunately causing long waits for our services and impacting the flow of patients through our hospitals."