Their lawyer argued "media coverage has been unrelenting" of the case which had unfairly linked her clients to a group of unruly tourists. She argued the two men were not part of the misbehaving group.
Anoushka Bloem made her arguments while attempting to keep the men's names suppressed.
She also claimed overseas media reports of the group of travellers may have already prejudiced her clients' case.
After verbal arguments were also made by the Herald, Newshub and the police, Judge Russell Collins declined the name suppression application.
Any alleged link to the group of tourists, which the judge said had gained "some notoriety", also did not satisfy him that publication of the defendant's names would compromise a potential trial.
However, Bloem indicated immediately that she would appeal Judge Collins' decision to the High Court - automatically imposing a 20 working day interim suppression order.
Both men also entered not guilty pleas and were remanded in custody.
Meanwhile, James Anthony Nolan, 26, remains wanted by police after absconding while on bail.
The British national, who is understood to also be part of the tourist group, has failed to appear at his court appearances over the last few weeks to face allegations of fraud, reckless driving and assault with a weapon.
The assault and driving charges are the result of an incident on Auckland's Takapuna Beach last month.
Krista Curnow was at Takapuna Beach on January 13 when the alleged driving incident unfolded
She said a car veered towards her and the British driver tried to take her cellphone while she tried to take a picture of the car's licence plate.
Police are also attempting to locate British nationals Tommy Ward, 26, and William Donohue, 25.
A family of British tourists first made headlines around the world after a video emerged of rubbish being strewn on the Takapuna Beach reserve.
The incident was filmed and showed a young boy threatening to "knock your brains out" when challenged by locals.
Tina Maria Cash, 26, another member of the travellers, earlier admitted theft charges this month after she stole energy drinks, rope and sunglasses from an Auckland service station.
She was convicted and ordered to pay $55 in reparation.
Five members of the group were also served deportation liability notices.