This answer on S.E's "the Workplace" says
The employment agreement might prohibit disclosing the names of specific clients.
To what extent is that legally enforceable?
I can understand that I may not be allowed to say on my CV/resume that I worked for certain government departments, but how is it in general? Further, I can see how a government department can, and can require the agency to pass that requirement on, but I do not believe that such a clause by an agency would hold up, as it is liable to deprive the signer of the the means of earning a living.
Can Agency, Inc really take legal action if my CV and my statements at interview say that I worked at (not for) BigCompany, Inc, on a contract basis?
If you want to keep you conscience clear(ish), you could always say "I work at BigCompany, Inc" and let others draw their own conclusions.
I will enlarge a little.
I have been a software contractor for much longer than you have been alive, and use "I work for" since BigCompany gains from my work (more than the middle-man); they use the software that I develop, while the agent only takes a cut.
But, the important part here is where you ask
Now, the question is, am I wrong in writing on my resume that I'm a consultant for BigCompany?
You could do that.
2017 - 2109 BigCompany, Inc, Position: consultant
Personally, I put
2017 - 2109 BigCompany, Inc, Position: senior software engineer
But I would never, ever, ever name an agency on my CV.
The reason is that hiring managers in the industry can gauge your suitability from companies where you have worked. So, having experience at British Aerospace is of interest to a hiring manager at Boeing, etc
It may also be worth mentioning that I am security cleared in Europe and work on sensitive government/military projects, and find risible the idea that a potential employer not be allowed to know of my employment history.
Tl;dr - can an employment agency enforce a clause forbidding those who work for it disclosing for which companies they have worked?
I have added aunited-states tag, as that is where the original question is based.