How can an employee drop the health insurance provided by the employer?

Under most circumstances you can drop coverage at open enrollment. You can check with your benefits administrator to see if there are any qualifying events that may allow you to drop earlier. Also, in some circumstances you can not drop coverage even if you want to. Here in CA if the employer's health insurance contract calls for the employer to pay 100% of the employees premium the employee can not decline coverage. Very often you will see a contract written at 99% even though the employer actually pays all of it just for that reason.