Yes, you should budget some amount of your emergency fund for healthcare expenses. How much you budget is really dependent on your particular anticipated costs. Be aware that health insurance likely costs significantly more than your employer charges you for access to its plan.
Since healthcare reform mandated guaranteed issue individual coverage you will have the ability to buy individual coverage for you and, if applicable, your family. When buying individual coverage you have essentially two choices, your decision hinges on whether or not you'd qualify for a premium subsidy. If your AGI is below 400% of the poverty line you'll be able to receive subsidized coverage at a state or federal health insurance exchange. If the subsidy is not meaningful to you, or you wouldn't qualify, you can buy an "off exchange" plan offered either directly through a carrier or an insurance agent (some insurance agents are also licensed to sell exchange plans though it's somewhat rare). In order to receive subsidized coverage you must buy through a state or federal exchange, or an agent licensed to sell exchange products specifically.
If your employer was large enough to be required to offer its plan via COBRA or you live in a state that extends the COBRA requirement to smaller businesses, you can choose that as well. Bear in mind this option is likely to be expensive relative to individual plans. It's becoming a less relevant solution with the advent of guaranteed issue individual coverage. COBRA is not a special type of insurance, it's a mandate that your employer allow you to remain on its plan but pay the full gross premium plus an up to 2% (10% for calCOBRA) administrative fee.
Despide popular vernacular, there is no such thing as Obamacare or ACA coverage. Obamacare reshaped the insurance market. The ACA outlines certain minimum coverage requirements, generally referred to as "Minimum Essential Coverage." While employers and plans are not "required" to meet all of these coverage requirements there is a penalty associated with non-compliance. The single exception to this is grandfathered plans which can still sidestep a few of the requirements. The penalty is harsh enough that it's not worth the cost of offering a non-compliant plan.
Whether you buy coverage through a state or federal exchange, through an insurance agent, or via your employer's COBRA program you will have "ACA" coverage (unless on the off chance your employer's plan doesn't check the "Minimum Essential Coverage" box). So generally all plans available to you will have $0 preventive coverage, pregnancy benefits, cancer treatment benefits etc.
Another thing to consider is your entire family doesn't need to be on the same plan. If your family is healthy with the exception of one child, you can purchase $0 deductible coverage for the one child and higher deductible more catastrophic plan for the remainder of your family. In fact you could choose COBRA for one child and purchase individual coverage for the remainder of the family.
The things to consider when you face a lay-off:
I tried to mitigate my use of "all" and "always" because there are some narrow exceptions to these requirements, such as the "Hobby Lobby" decision allowing closely held organizations with highly religious owners the ability to remove certain contraception benefits. Understand that these exceptions are rare and not available to individual plans.
The guideline for the size of an emergency fund is just a guideline. I've usually heard it expressed as "3 to 6 months," but everyone has a different idea of exactly how big it should be.
The purpose of the fund is to give you enough cash to be able to pay for unexpected expenses that have come up that you have not budgeted for without you having to borrow money to pay for them.
To figure out how big this fund should be, we look at the worst case scenario. Suppose that you lost your job tomorrow. What would you do?
Cut your expenses. You'd probably be much more careful how you spend money.
Secure health insurance. This would be done by either continuing your employer's policy with COBRA, or by purchasing your own insurance, likely through the Obamacare/ACA market. Keep in mind that most likely your employer is paying for a portion of your insurance now, so this expense will go up quite a bit no matter which option you choose.
Look for another job. You'd probably begin your search for a new job immediately.
The size of your emergency fund determines how long you will be able to go without income before you need to start a new job.
Regarding cutting your expenses, it is up to you how much you would cut. There are things that are easy to cut temporarily (or permanently), such as restaurants, entertainment expenses, vacations, etc. You would probably stop retirement investing until you have income again. The more you cut, the longer your emergency fund would last. Things you don't want to cut are necessities, like housing, groceries, utilities, transportation, etc. I would also include health insurance in this list. Certainly, if you have a pre-existing condition, you do not want to let your health insurance coverage lapse.
Your employability is also a factor. If you believe that you would have an easy time finding similar employment to what you have now, your emergency fund might not need to be quite as big as someone who believes they would have a harder time finding another job.
Yes factor into your fund the cost of health insurance.
You basically have three options when facing a loss of income for 3-6 months:
Pre-ACA the COBRA one was the default option many planned for because there was no need to change doctors. Of course many people were shocked how expensive it was compared to just looking at the employees share of the monthly premium.
For planning you can do some research into the cost of one of the ACA approved plans in your state. Keeping in mind that the lack of income might qualify you for a subsidy.
As to the coverage level, that would depend on your situation and the perceived gap. I have known many people who didn't have to pick COBRA until after the new job started so they knew exactly what they needed to cover and what their bills were during the gap.