Will my child's juvenile expunged record show up in a background investigation for a Government job?

Depends on the level of the clearance needed. Most likely, yes.

I will agree, are you going to be a janitor at a army recruiting office or an computer program in the Pentagon. When I worked for the Federal Government, they sent people, to talk to my some of my high school teachers, neighbors where I used to live, and a lot more.

The question, normally on application, is where you ever arrested or have you ever been convicted.