You have certain rights when you are leaving a job which can help protect your health insurance coverage now, or later.
HIPAA: If you participated in your employer’s medical plan, the plan usually must give you a certificate of health coverage after your employment ends. You may need this certificate when you start your new job, in order to avoid health benefit restrictions at your new employer. So be sure to ask for this certificate as soon as possible, when you lose coverage under the old plan or your COBRA continuation coverage ends, or when you start your new job.
COBRA - there are Federal and state protections which let you pay to keep your current health insurance.
If you had already had a right to payment or reimbursement under your medical plan before you left your job, the plan still has to pay even after your job ended. Make sure that you do everything you have to do to get reimbursement (or payment to your doctor), and file an appeal immediately if the plan doesn't pay (often, health insurnace companies refuse to pay valid bills, just to see if you will appeal it). If you're not sure of the rules under your plan, ask for the Summary Plan Description from HR or the Personnel Office.
What can you do if you can't afford health insurance?
First of all, don't tell the plan or your boss that you are not going to sign up for COBRA until the deadline. If you had a big medical emergency, you could sign up (and pay) and you would be covered from the last day of work.
If you are leaving a job with insurance, get as much medical care - annual check-ups, tests, proceedures, eye exams, and fill prescriptions -- as possible before your coverage ends.
Can you sign on to your husband, wife, or domestic partner's plan (some plans permit domestic partners' coverage)? You have 30 days of special enrollment rights to sign up after you lose your benefits. Many people get married just to get insurance coverage (single payer health insurance would be a great protector of the sanctity of marriage).
Find out what plans your state has for people who can't afford individual insurance. Try to cover your kids through your State Children's Health Insurance Program (SCHIP). Some states have special plans for low-wage or unemployed workers, as well as plans for people with pre-existing conditions which make it hard to find a health plan that will take them.