Suppose a company has a disgruntled employee. The employee starts badmouthing them all over the place. Assume it doesn't rise to the level of libel or slander, but they're still causing significant problems. Is the company supposed to be required to keep employing this person or face a freedom-of-speech lawsuit? That seems absurd.

In this case you document the incidents. You establish a pattern and attempt to demonstrate intent or reckless disregard with regards to the impact of speech on the company and its business, in quantifiable terms if possible. You appeal to previously established, legally valid agreements made between the employer and the employee on what sort of speech restrictions are expected. And then when you fire the person, you tell the truth. You explain exactly why the relationship was terminated, how their exercise of free speech damages the company, and how they themselves are responsible for it.

The trick is really when it comes to actually proving that an employee is "causing significant problems" when their actions cannot actually be classified as libel, slander, or some form of harassment.

The fundamental problem here I think is the assumption that an employer is somehow "responsible" for the employee's livelihood. Employment is a simple business transaction.

No, the fundamental assumption is that any actor has a responsibility to uphold business agreements in an honest and ethical manner. This includes not using the threat of termination as leverage to suppress exercise of 1st amendment rights. That includes making reasonable attempt to respond to speech with speech before taking action, and ONLY escalating to disruptive action against the speaker when there is a clear and compelling reason to do so.

I'm all for at-will employment in principle. But the story doesn't end there. Just because you "can sometimes be justified firing someone for things they say" doesn't mean it's always the right thing to do. You have to look at each situation to make a reasonable evaluation. In this case, according to Lynn, Google applied economic pressure to (supposedly nonpartisan) New America to coerce them into into punishing Barry Lynn for praising an EU decision that hurt Google.

Lynn was not an employee of Google. Lynn was doing his job, the way he was supposed to do his job, in a principled manner which he claims is consistent with how he'd done it for the past 15 years. He and his division were employed through a 3rd party organization of which Google was simply one funder. Google, rather than responding to Lynn with a counter argument or some other speech-oriented response, instead abused their patron relationship with his employer to punish him for what he said.

Whether or not New America had "responsibility for the livelihood" of the employees is entirely beside the point. The point is to address the accusation of Google engaging in a wholly inappropriate bullying behavior. It appears they set out with intent punish someone for valid political expression using a coercive method (threatening the speaker's employer). That is the general principle at stake here. Responding to free speech with coercive action is by default unethical.