I have no problem with keeping the details of an investigation classified. The issues is that you can't even say there is an investigation.

That would be less than a $100 million per year operating expendeture.

Which Yahoo shareholders would be irate about, and the Yahoo corporation could not trivially afford to throw away regardless. That would have destroyed about 1/4 of Yahoo's profit at the time.

That would be bad enough, however the next step by the government would be to increase the fine amount, were Yahoo to hold out over time. Kick it up to $1 million per day and Yahoo would have folded no matter what. The problem is pretty straight forward: the government could push it as high as necessary to compel the result they wanted.

It is also a straight forward violation of the first admentment.

I doubt that.

"Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances."

talks about Congress, not an executive branch. the latter can gag a person, or a company.

The First Amendment applies to all three branches of the government, not just Congress. The executive branch can attempt to obtain a gag order, but the 1st Amendment nonetheless places limits upon them.

See, for example, https://en.wikipedia.org/wiki/New_York_Times_Co._v._United_S...