I can speak from the Army side. I received a full ride Reserve Officer Training Corps (ROTC) scholarship in Engineering. I Have a B.S. in Construction Administration (its like Civil Eng and Business).
As Vandium pointed out - especially for grad school generally you have to work for the Army for a few years before you will be able to ask the Army to pay you to go to Grad school.
For example, in mid-career as an Officer (7-10 years), you can apply to do exactly that. As you can imagine it is competitive, and the metric is your performance as an Officer in the Army, not "how academically smart you are." There are some newer incentives right from ROTC but they appear similar. Of course if you get one you will have to stay on Active duty for a period of time after (not usually less than 4 years). main reason for this incentive is to keep you on active duty till the 15 year mark, which then it would be "not smart" to get out after 15 years, because if you stay for another 5 years you can retire with a pension (which as an officer is not shabby at all). That is Active Duty only. There are military post graduate opportunities (Navy Post Graduate School come to mind) which one can apply for at any time. I know an Engineer Officer that got one of those, he enjoyed it. Don't expect leaving the Army anytime soon after one of those tho :D
In the Reserves or National Guard you don't really have this option available, (Part time status - 1-2 weekends per months a couple weeks a year - and/or deployments of 1-2 years overseas).
In the Reserves or National guard, however, there is the new GI Bill which can be used for anything (Undergrad, Grad, PhD), pays tuition, 1000 bucks for books, tutors and you get ~$1100 bucks per months to pay the bills for whatever. The catch of course is for every month of benefit you have to have been deployed in the Army (in support of the war) for a month up to 36 months, (maximum 48 for some special circumstances). With no wartime service you can get around $200 per month and some States will pay for the tution 100%.
Without knowing anything about you, I would recommend looking at the Dept of Defense program: http://www.asee.org/fellowships/smart/ . Of course that is an all-or-nothing application.
It's probably mega-competitive but does not require military service. The Army does not do research internally, it is done by contractors, and/or DoD civilians. There are jobs where as an Officer, you may manage a project for the Army being done by civilians or contractors.
Remember, the Army doesn't make / research / engineer new bombs / vehicles / weapons etc- US Private industry does, then sells it to the DoD under contract. The Army does build things (Civil, some mechanical), a lot of times the actual building and design is outsourced though and as an Officer you act as Project Manager.
One thing you DO get as an Officer, especially as a Combat Arms Officer, fighting rapidly develops your problem solving ability, creativity, leadership, project management, and how to get funding for projects from the government.
