The common wisdom regarding self employment and health insurance is that if there's likely to be any health related issues in your future you should think long and hard about self employment. Health insurance can be quite expensive, though there are ways to keep your expenses down enough to make it possible to have it all.
For starters, the most common route is to have one partner continue working for an employer that offers good quality family benefits, while the other person peruses their dreams of self employment. Health insurance premiums for family rates (versus single person rates) are sometimes quite reasonable, depending entirely upon the sort of contributions that employers make and the sort of group rate they're able to offer. It will certainly be less than taking out two policies.
If you already have a good policy, you may prefer to keep your own policy for a time after you leave your job for self employment. Health insurance laws such as the COBRA act of 1986 require you to be given access to the same group rates (though not with your former employers contribution) for a year and a half. This doesn't apply if you were fired or worked for an employer with fewer than 20 people. In the latter case, you may be able to actually get a better rate as part of a group of self employed people.
Even if you're a single parent, there's no reason you have to go it alone all the way. There are several organizations of the self employed who pool their resources together to get better insurance rates for their membership. Though these rates are not often as good as those large companies are able to get, it can represent a significant savings over the rates that you'd pay as a single person seeking family coverage.
Of course, a great many people choose not to have health insurance, as single people, but having a baby or raising one is sure to see you in the doctor's office somewhat regularly. As such, you might want to consider a health savings account or supplemental insurance to cover the cost of copays and deductibles.
Some states offer a type of public assistance that covers the cost of basic health care for children, though being self employed, you may not be eligible. Check with your state to find out if such a program exists in your area.
It is vitally important that you keep the costs in mind when contemplating starting a family and a business at the same time. As one of the newly self employed, health insurance is also the largest expense you'll encounter when first starting up. You may find yourself making a choice.