Of course they want to block the book, as they are sure it will not paint them in the way they want to be seen.
As Wally626 mentioned, the tactic they are using to block it is not a First Amendment issue. It is a contractual issue, involving conflict of interest and intellectual property rights. It just looks like a 1st Amendment issue on the surface.
Nowdays, when you work for anyone above the mom & pop business level, there is almost always some kind of conflict of interest/intellectual property policy or agreement involved. Abiding by it is a condition of employment.
Anything that you do outside of your job duties that could be, or could create the appearance of a conflict of interest must be reviewed and approved by the company before you do it. Otherwise the company can take legal action against you for doing it. Most of the time, they can also terminate your employment for doing it, aside from any other actions they could take.
Say I worked for Westinghouse (for example), making appliances. I want to write a book about the history of the US Marine battles on Guadalcanal during WWII. Even though it has NO relationship to my employer I still have to get their approval (because of the employment contract I work under) before I can publish it. Otherwise they can take action against me.
If I wanted to write a book about the ATF, I could, and would not need to get their approval. I don't work for them.
But if I did work for them, I would have to get their approval. No matter what the subject matter was.
While it is a free speech issue, it is NOT a 1st Amendment free speech issue. ITs a contractual matter that he voluntarily entered into, when he took the job.
On the surface, its a simple solution, terminate employment with the ATF, and then publish. That will knock the legs out of their argument.
IF he does that, I think they will still try to block the book, but they will have to use a different argument. Essentially they would have to prove that he used classified, sensitive, or proprietary information, without authorization. OR that he used company (in this case govt) resources in writing the book.
Better hope he never had so much as one page of draft on any govt computer...otherwise, they have standing for that argument.
All else being equal (and it almost never is) bigger bullets tend to work better.