Filling in the EOI is an application for Residence under the Skilled Migrant Category. Though you are on the application as a secondary applicant, it is ONLY your husband's qualifications and experience that will be being investigated. As long as he qualifies and is granted Residence, you get it too, regardless of any qualifications you have or don't have. That gives you the right to work at anything you want, skilled or unskilled, or run a business, or do nothing, or be a student at domestic rates (only you won't be eligible for any NZ grants until you have been resident for three years).
If your husband's new employer wants him to be in NZ to start working more quickly than the SMC application can go through, he can apply for a temporary work visa
http://www.immigration.govt.nz/migra.../requirements/ or WTR visa
http://www.immigration.govt.nz/migra...ce/default.htm. Either of these can be processed more quickly than the SMC and can get him there and working while the residence application is ticking along in the background, taking the time that it takes.. He may find the employer is already accredited, and all ready to sponsor him for one or other of the work visas. Both of these visas give your husband the right to sponsor you for a partner-sponsored temporary work visa
http://www.immigration.govt.nz/migra...milystream.htm, which also allows you to work at any job. You can send both applications at the same time, linking them with a cover letter, and the CO will work the whole thing as one case, supplying missing information like his visa number as it becomes available.