The best source, unfortunately, does not explicitly distinguish between nonprofit, for-profit, and government employers. However, it does provide detailed listings of occupations and industries that can be used to estimate the likely distribution of employees in many segments of the nonprofit sector by gender and ethnicity.
The Bureau of Labor Statistics data on ethnicity and gender by industry is derived from the Current Population Survey: http://www.bls.gov/cps/cpsaat18.htm. It includes detailed industries such as hospitals, higher ed, religious congregations, membership associations, and the arts. The raw CPS data COULD be analyzed to distinguish for-profit, government and nonprofit employment but there are concerns about undercounting nonprofits in this survey.
Tables by occupation are also available at: http://www.bls.gov/cps/cpsaat11.pdf
Other tables on gender are work are at: http://www.bls.gov/cps/wlf-databook2011.htm
For information on the ethnic composition of boards, see Francie Ostrower’s report on diversity: http://www.urban.org/publications/411479.html