The advertisement was taken down shortly afterwards, but said the business "will not be discriminating either whilst employing or serving our guests".
Lone Star's head office also issued a statement saying all franchisees were "required by the terms of their franchise arrangements to comply with all relevant laws, legislation, and government requirements, including health requirements".
Now a Facebook post has been put on community pages asking people to "Help businesses say no to segregation".
It said people should support Lone Star New Lynn by bringing family, friends, a supportive sign and a picnic blanket, before ordering food from the restaurant and eating it outside its premises.
The post said Lone Star New Lynn is aware the supporters will be coming and loved the idea.
The Herald understands this is not correct and that Lone Star's management is not aware of the planned picnic.
Owner Brendan Pascoe said he could not provide comment to the Herald.
The new post also follows on from another yesterday that claimed an incident involving police had taken place at the Lone Star New Lynn store.
The Herald also understands this did not take place.
It comes as Aucklanders got their freedom yesterday with bars, cafes and restaurants opening again for customers to eat and drink indoors.
The strict lockdown preventing hospitality venues from opening ended yesterday after 107 days in the city of 1.7 million people.
But while cafes, bars, restaurants and gyms are among businesses required to ask customers for vaccine passes if they want to operate as normal, other essential services are not allowed to ask for vaccine passes.
They include:
• supermarkets
• pharmacies
• all health and disability services
• social services
• food banks
• petrol stations
• public transport (excluding air travel)
