The nine-team tournament will be staged in Qatar from March 14-30 to decide which team advances to a playoff against the fourth-placed side from CONCACAF, which features nations from North and Central America and the Caribbean.
World governing body FIFA has given the Oceania nations a one-day extension to the Mar. 21-29 international window but they will have to haggle with clubs to release players during the first week of the tournament.
New Zealand is disproportionately impacted by the scheduling given it has more players at professional clubs than their Pacific rivals.
"This would happen nowhere else in the world," Hay told New Zealand media. "I'm here in Denmark at the moment and trying to explain how this has transpired; nobody can get their head around it.
"I spoke with the Danish Football Federation today, and they were just bamboozled.
"Imagine saying to the Danish national team that you've got World Cup qualifying but you can't pick your best players.
"For me and our players, it's highly disappointing."
New Zealand were the last Oceania team to qualify for the World Cup, having made the 2010 finals in South Africa.
"Every single player that I've called up to date is a professional player, they're operating in professional environments, but clubs aren't going to be willing to hand over their players in the middle of the season," Hay said.
"It's a tough one to swallow, that we are not going to have the opportunity to bring our best players together to give us the best opportunity to qualify."