You have to talk to the payroll department and find out what they are going to do. If they are not going to withhold CA tax there is no point in sending them a CA withholding form. If they are going to withhold CA tax, the form you have to give them is a DE 4, not W-4. I suspect that the company might not have any other CA employees, and might not be set up to withhold CA tax.
You cannot claim Exempt for NJ because you do not meet the requirements to be exempt. You are not exempt from NJ tax on income that you earn in NJ.
Technically your employer should keep track of the number of days that you work in each state, and the state wages in box 16 of your W-2 at the end of the year should show the amount that you earned in each state. (You should keep track too, to check that they get it right.) There should be two lines of state information on your W-2, one line for CA and one line for NJ. Check with the payroll department and see if they are going to do this by the book. If you're only going to be in NJ for a few days, they might prefer to ignore it and report all of your income as CA wages. For a matter of a few days, you could go along with that and not file a NJ tax return. A lot of companies don't bother with the extra paperwork when the amounts involved are small. If they are not going to report any NJ wages or withhold any NJ tax, there is no need to give them a NJ withholding form.
The main thing you want to be sure of is that they do not report all of your pay as NJ income or withhold NJ tax on all of your pay. You can live with any other fudging that they want to do, but having your CA income reported as NJ income would be a problem. Make sure payroll clearly understands that you are a CA employee, not a NJ employee.
