"The union did not organise the strike action by BRT workers. Workers took it on themselves to go on this strike action," Samwu spokesman Tahir Sema said.
"It is illegal and it is not protected. They would stand the risk of losing their jobs," he said, denying it was a leverage ploy ahead of the imminent start of the soccer World Cup.
The buses, which were introduced last year, were to have been an integral part of transport plans for fans travelling to stadiums which would host the opening and closing matches and 15 other games.
Sema said shop stewards would speak to the drivers on Monday to tell them to go back to work, but at the same time the union would step up talks with the company to address their members' concerns.
A strike is considered illegal if workers have not been through a lengthy process of mediating a settlement to a dispute.
The drivers want the to company recognise their union and let them negotiate in the municipal bargaining chamber, which the company does not agree with.
Rea Vaya interim chief executive officer Jackie Huntley said the company felt because the enterprise is privately owned, and not owned by the municipality, they would be in the wrong bargaining chamber for salaries and other the labour-related issues as they are not municipal workers.
She said the drivers also want to become permanent employees from June 1, instead of July 1 as expected, but this was not possible.
But Sema disagreed, saying the buses, which form part of the government's Bus Rapid Transit system, use many municipal resources and infrastructure, and the drivers are entitled to be represented by Samwu, and not another union suggested by Rea Vaya.
A court judgment on this matter is awaited.
Since its rollout in Johannesburg last year, the system has lurched through protests by taxi operators concerned about income losses, and some of the buses being shot at.
The Rea Vaya strike also takes away one of the few transport options left to commuters affected by an SA Transport and Allied Workers' union strike which has stopped Metrorail from running.
At the beginning of May one man was killed and eight injured when a bus was shot at. One man is expected to appear in court on Monday in connection with the shooting.