Any job where you can gain practical experience is a good job.
The benefits of working on the private side (corps such as Boeing) is you will probably make more money, have greater mobility, quality training, and more opportunities to advance your career withing the company.
The advantages of a government job include more security, good benefits, and making a contribution to society. Also, government employment can lead to good jobs in the private sector later on if you decide to change.
Connections are always helpful, but are about equal as a barrier to entry in both sectors. Without connections, you need to play the numbers game - apply to a lot of positions that seem interesting to you. One tip - with all the applications you submit, follow up relentlessly.