I've got some insights to share with you on how you can save your employees data with New Jersey information.
The selection of the states on the W-4 info will depend on the employee's address that you've entered. To get this issue sorted out, I suggest verifying the employee details to see if everything is correct.
Once verified and the problem continues, I recommend accessing your account on a private window. This helps us verify if the issue has something to do with your browser.
QuickBooks Online collect cache files to speed up your browsing. However, too much may cause unexpected behavior like being unable to save employees' data with NJ.
If you're able to save the employees' info successfully, you'll need to clear the cache of your regular browser to prevent this hurdle from happening again.
On the other hand, I've got here some articles that you can visit for more details about adding an employee and managing payroll in QBO: