This is a style question, so there is no right answer unless it be that whatever style preference a publishing house insists on is the right answer. Still, there does seem to be a tendency in mainstream U.S. style guides to treat military units (companies, battalions, etc.) as part of a proper name when it appears with a particular identifying word, letter, or number designation.
For example, although Words into Type, third edition (1974), doesn't show any awareness of the Alpha, Bravo, Charlie, Delta alternative designations for companies, it has this general style advice for dealing with military units:
Organizations. Cap the names of organizations of the military. Army, navy, and air force, used as noun and adjective, are not usually capped (army post, navy patrol, air force blue), unless they refer to the official organization of a country (Army policy, Navy officer) or if the name of the country precedes (United States Army, British Navy).
[Relevant examples of unit names:] B Company, Company B
The Associated Press Stylebook (2002) doesn't address the OP's specific question, but it strongly suggests that it, like Word into Type, would favor capitalizing company in the phrase "Delta Company" but lowercasing it when cited generically as "the company":
military units Use Arabic figures and capitalize the key words when linked with the figures: 1st Infantry Division (or the 1st Division), 5th Battalion, 395th Field Artillery, 7th Fleet.
But: the division, the battalion, the artillery, the fleet.
The Chicago Manual of Style, fifteenth edition (2003) takes the same general approach as AP:
8.120 Armies, battalions, and such. Titles of armies, navies, air forces, fleets, regiments, battalions, companies, corps, and so forth are capitalized. Unofficial but well-known names, such as Green Berets, are also capitalized. Words such as army and navy are are lowercased when standing alone, when used collectively, or when not part of an official title.
So the consensus does not favor your editor's view that "Delta company" is preferable to "Delta Company"—but as choster observes in a comment above, that fact won't help you if the editor is enforcing a house style decision that is graven in stone.