but Anna Faris is, but only in my dreams.

Stan's McClatchy link spells out the problem - parsing "supporting"; for which, legislative history is sparse:

Sen. Jeff Sessions, R-Ala., the author of the 2004 amendment and of the original MEJA, said simply on the Senate floor June 16, 2004, that "this amendment would give the Justice Department authority to prosecute civilian contractors employed not only by the Department of Defense but by any federal agency that is supporting the American military mission overseas."
The indictments (linked above - p.1 of this thread) emphasize this:

Ridgeway

At all times relevant to this Information:

1. On or about September 16,2007, defendant JEREMY P. RIDGEWAY was employed by the Armed Forces outside the United States, as defined in 18 U.S.C. § 3267(1), that is:

a. The defendant was an employee and a subcontractor of Blackwater Worldwide, a company contracting with the United States Department of State, who was employed to provide personal security services in the Republic of Iraq, which employment related to supporting the mission of the United States Department of Defense in the Republic of Iraq.
and the Others

At all times relevant to this Indictment:

1. On or about September 16,2007, defendants PAUL ALVIN SLOUGH, NICHOLAS ABRAM SLATTEN, EVAN SHAWN LIBERTY, DUSTIN LAURENT HEARD, and DONALD WAYNE BALL, were employed by the Armed Forces outside the United States, as defined in 18 U.S.C. §3267(1), that is:

a. The defendants were employees and subcontractors of Black water Worldwide, a company contracting with the United States Department of State, who were employed to provide personal security services in the Republic of Iraq, which employment related to supporting the mission of the United States Department of Defense in the Republic of Iraq.
The use of "employed by the Armed Forces" in the first paragraph has caused confusion here and elsewhere - because they weren't employed by the Armed Forces in the general meaning of that term.

We'll see what Judge Urbina will do with this one. His decision in the Uighur cases is no precedent here. In fact, there is no precedent here.