From a white-paper titled Selected Characteristics of Private and Public Sector Workers:
Occupations sorted by size. The numbers shown are 1000s:
To answer the question in the title: Probably not. Cops, military and firefighters are probably more conservative than the US average and they have the very highest ratio of public-to-private employment.
Education and community and social scientists have the next two highest ratios and they tend to be liberal.
On the other end of the chart, farmers etc tend to be conservative while production tends to be liberal (especially those that are unionized). Sales is a mix...successful sales people on commission tend to be conservative while the lowest levels of employees (often minimum wage/part time) tend to be liberal.
Full chart here: