If private armies are subject to international laws, or US laws regarding them, then they should be allowed to be sent. The private armies are often well equipped and experienced, often former military members themselves. The real danger is because the private armies are often not held accountable for laws broken.
Yes, American companies should be permitted to form and deploy private military in war times, as long as the private military works with the United States to further the objective of the war. If the United States can meet its military objectives by using private contractors, it should, by all means, do so, if it can be done at a cost-saving to the American taxpayer. A person should carefully consider joining such an army, as they might not have the same international protections as official government military actions.
Corporations should not be able to have their own militaries in times of war. If there was a serious war, there would need to be tactical and coordinated efforts in order to be successful. You don't want every Tom and Jane going crazy with guns trying to do their own thing.
Companies should not be given that type of power. If you give a company the ability to form their own private army/military than they will eventually use it against other people of this nation. That would cause other people to need to get a military out and start a vicious cycle.
I do not understand why it would be necessary for American companies to deploy private militaries in times of war. I do not understand why American companies, that are in business for profit, would need to do such a thing. I also find it difficult to believe that they would find employees who would be willing to participate in such a thing.