"We didn't come here for the free health care because we already had that in Germany, where in fact it's a better health care," said Mrs Mahon.
"We made the decision to come to New Zealand because it is our home, or we thought of it as our home and we love it here."
Mr Mahon, who also holds German citizenship, has been a New Zealand citizen since birth after his father applied for it.
He first moved to New Zealand in 2004, but returned to Germany to work in 2012, and had always intended to return here.
Liam does not qualify for New Zealand citizenship because Mr Mahon became a citizen by descent, and this cannot be passed on to his children.
He has been issued an interim visa while his case is being considered.
Immigration New Zealand Area Manager Michael Carley said: "If INZ is not initially satisfied that an applicant has an acceptable standard of health, we must refer the matter for assessment to an independent medical assessor for their opinion.
"People not in good health may be granted a medical waiver if the person meets all the other requirements for approval of their application."
Mr and Mrs Mahon met in Auckland in early 2007 when she was working as a writer on a working visa and him as a design engineer and got married on Waiheke Island at the end of 2008.
They have thought of NZ as their home ever since, but being an expat family meant they've moved back to both countries of their birth to spend time with their families.