Other countries have compulsory military service, for example Finland. Generally speaking I am a fan of the idea that everyone should be required to do some kind of community service for 2 years once they turn 18. Military service would just be one option, could be other kinds of civic engagement. This can really help people feel connected to their society and understand that there is something to show up for.
BUT - this really only works if there's a social contract in place. In the United States it's hard to see how compulsory service works if people don't feel like the country is showing up for them.
These days, what are American soldiers dying for? A society with great health care? Fantastic education? Wealth and social stability? Absolutely not! Until that changes I don't see any good reason why we should send our young people off to die. (EDIT: if you want a sobering experience, visit a military graveyard and pay special attention to the ages of the soldiers. We might as well call a military graveyard a children's graveyard).
And to agree with others on this thread, the folks who push for war should 100% be required to participate in them and lead from the front. Don't sell the rest of our lives while you hide in a nice air conditioned bunker.