Mr Shaw from Wellington and Mr Hague from Greymouth have put their hats in the ring to replace co-leader Russel Norman who announced his resignation as leader in January.
Both leadership candidates agreed that if they won the role, Mr Norman would be leaving big shoes to fill.
"Both of us do have the advantage of being more experienced than Russel was when he started in the leadership role," said Mr Hague.
Mr Hague is the party spokesman on housing, health, sport and recreation, rainbow, alcohol and other drugs.
He has held senior positions in the health sector, including directorship of the Aids Foundation and is a former chief executive of the West Coast District Health Board.
He was first elected to parliament in 2008 and has been an active campaigner against racial discrimination and a champion of gay rights, the Treaty of Waitangi, cycling and the families of the Pike River disaster.
"Housing, health and household incomes are all inextricably linked and healthy housing is a priority for me - the long-term cost of poor housing is poor health."
Mr Shaw said he joined the Green Party as a 16-year-old before going overseas and living in London for 10 years where he worked for business consultants PricewaterhouseCoopers.
Elected to parliament in 2014, Mr Shaw is the party spokesman on economic development, trade and investment, business, open government, electoral reform and state-owned enterprises.
"I am proud that the Green Party has put child poverty at the centre of concerns for New Zealanders - no party is denying the problem now.
"Child poverty and water quality are the biggest priorities that we need to address right now."
The other two contenders standing for leadership are Gareth Hughes, party spokesman on broadcasting, energy, mining, research and tertiary education and Vernon Tava, a Green Party co-convenor in Auckland who is not currently an MP.
Metiria Turei is the sole nomination to be re-elected as the Green Party female co-leader, a position she has held since 2009.
Mr Shaw and Mr Hague say the co-leadership model works well for the party and two leaders bring a good mix of abilities to the roles.
The new male co-leader will be named after voting at the party's annual meeting on Saturday, May 30.