Bosses at Marsh Ltd have identified the pair and have started an investigation.
Sources told NZME. News Service yesterday that the man holds a senior management position at the firm.
He was not answering his work mobile yesterday and nobody was at the family home.
His wife was said to be distraught over the incident.
"She had no idea until she came across it on Facebook on Monday night. She's in pieces and can't even speak to him," a source said.
The female employee, understood to be in her 20s and recently engaged, has not responded to messages.
A Marsh employee said neither worker was at their desk yesterday.
The mood at the Christchurch branch is said to be "rather tense".
Senior Marsh officials are believed to have flown down from Auckland.
"The other employees are all keeping their heads down and are just trying to get on with things. It's a pretty surreal situation for them," an insurance industry insider said.
Employment law experts say the couple could lose their jobs over the incident, and Privacy Commissioner John Edwards told Radio New Zealand that it was unlikely they would be able to sue for breach of privacy.
Marsh chief executive Grant Milne did not return calls yesterday.
He said on Monday that the company was treating the incident "very seriously", but refused to say if the pair had been suspended.
There has been some backlash against those who filmed the incident, and several commentators claimed the couple's privacy had been breached.
But Mr Edwards said it was unlikely the police could take action if a complaint was made.
"The people who were watching, we may criticise them for a lapse of decency in taking advantage of that, but it is unlikely there'd be any legal liability for their action," he said.
Employment law expert Blair Scotland said the pair could lose their jobs. The key element was whether Marsh believed it had been brought into disrepute.