The Census Bureau provides this type of information. However, the amount of specificity you are asking about is more than what they are willing to provide (for privacy reasons). The closest table is B27015 which covers general insurance status by household income category at the Census Tract level. Oh, and for reference, the Census Tract level is the lowest geography that will be available, and with any estimation at a small geography, make sure the margin of error is reasonable.
If you want specific counts of the population with a specific type of health insurance, you have options which include: Employer-Based, Direct-Purchase, Medicare alone, Medicaid alone, Tricare/Military, and VA. You could even try a table which examines the population of people with discrete combinations of health insurances.
If you are interested there are geodatabases (ArcGIS 10.1 or Later) that have the information joined to shapefiles here.