If you have flexibility in determining your last day at the office, pick the start of the month. Many companies have policies where health insurance coverage lasts until the end of the month.
This may not be the case if you are paid hourly. Please check with your HR representative. Some companies may also charge you the monthly premium in your last paycheck. Many companies often pay for a large percentage of the total monthly premium, so this will still benefit you.
When your health coverage with the company does end, certain employees have the ability to extend it for another 18 months, though you have to pay both your employer part of the premium plus how much you were originally paying. This is called COBRA coverage.
There is a 60-day window from your last day to when you have to decide whether you are enrolling on COBRA coverage or not. But this decision is retroactive. So wait on electing for COBRA coverage until you have found a job, have not found a job but require health coverage, or have a medical need.