It is generally not recommended to mount kitchen island countertop supports directly to the cabinet back panel. The back panel of a cabinet is typically made of thin or medium-density fiberboard (MDF) or plywood, which may not provide sufficient structural support for the weight of a countertop.
Instead, it is advisable to mount countertop supports to the cabinet's side walls or the structural components within the cabinet itself, such as the frame or cross supports. These areas are typically designed to bear the weight of the countertop and provide the necessary stability.
If you're planning to install a kitchen island countertop or make any modifications, it's best to consult with a professional contractor or a kitchen designer. They can assess your specific situation, recommend appropriate support methods, and ensure that your countertop installation is safe and secure.