I'm active duty with Air Force JAG. I cannot speak in regard to the Army or Marine Corps but AF and Navy JAG are incredibly difficult to get into. The Direct Appointment selection rate for AF JAG is 7%-11% annually.
It is difficult to set out exactly what makes for a competitive JAG applicant. The JAG branches obviously look at different criteria in their applicants than BigLaw. Grades and all the other academic activities (law review, moot court, etc) matter. The rank of the law school does not seem to make that much of a difference. The two bases I have worked at have JAGs from T14 through T4. Being at a T14 will not write your ticket for you the way it will in BigLaw. No Staff Judge Advocate (SJA, the colonel that runs each base's legal office) will be impressed with the JD on your wall. The flipside is that if you are at a TTT, you are not out of the running. For consideration, I was a T25 grad, top 1/3rd of the class, moot court team, secondary journal editor.
Being a commissioned officer in any branch is an incredibly big deal and is always highly competitive. Don't fool yourself into thinking that enlisting in the Army is same as earning a commission.