Is the original statement above a sound argument: "if you think free speech is an attack then you have a problem with free speech"?
The prime method of philosophy is to use reason and language to examine relationships, truths, properties, etc. So, let's begin there. Let's take it piecewise, and examine the truth condition of each statement.
Criticizing the Press
"Criticizing the press is not an ‘attack’ because that would imply an action that would do harm such as: outlaw a publication, jail a reporter, etc.
Definition of attack at MW:
1 : to set upon or work against forcefully attack an enemy fortification
2 : to assail with unfriendly or bitter words a politician verbally attacked by critics
3 : to begin to affect or to act on injuriously plants attacked by aphids
4a : to set to work on attack a problem
This is false because criticizing the press certainly can be an 'attack' even if it does not result in a physical attack or harm to the organization or members as in 1. Consider definienda 2, 3, and 4a. One can certainly use unfriendly or bitter words. One can harm the reputation of a medium which would affect injuriously. One can consider it an attempt to reform the problems of an outlet. Any of those senses allow us to characterize the criticism of the press as an attack according to conventional word meaning.
Criticism can be considered a verbal attack
Calling out bias or false reporting (ie: fake news) is just expressing an opinion and free speech.
This statement has two predicates, one characterizing the action as expressing an opinion and the other expressing it as an act of free speech.
Certainly, criticizing an outlet for propagating falsehoods is an act of free speech, and is a perfect candidate as an example because it's likely to be the type of speech those in power particularly the government may not like if they control a news outlet. The difference between criticizing the Chinese state medium and the UK BBC is that the former can land you in trouble and the latter is encouraged. Therefore it is true that criticizing the various media is an act of free speech.
But what about it being an opinion? That is false. If a newspaper is printing falsehoods, deliberately or otherwise, calling those statements falsehoods is not an opinion. It is a matter of fact. And commenting on the reputation of that outlet with a statement pointing that out is also a matter of fact. It may be a true or false fact, but this isn't in the realm of subjective opinion.
Objecting to Free Speech
If you’re saying that ‘free speech’ is an ‘attack’ then you have a problem with free speech."
A good place to look for lines of argument against this proposition are my answer to the PhilSE post: Is the statement “If you are against criticizing the media, then you are against free speech” begging the question?
This proposition is open to many interpretations, so let's take one:
P1. Using free speech to criticize speech is an attack.
P2. (It's objectionable to attack.)
C. Therefore, one who attacks another's speech objects to free speech.
Well, that's just nonsense. The definition of free speech according to WP:
Freedom of speech2 is a principle that supports the freedom of an individual or a community to articulate their opinions and ideas without fear of retaliation, censorship, or legal sanction.
How can the exercise of free speech to object to falsehoods that may be a tool of repression be an attack on free speech itself? This is a typical response of authoritarian regimes and is Orwellian doublespeak. There's a clear contradiction here. What fallacy is at play? I would argue that according to Damer's taxonomy in his Attacking Faulty Reason, this constitutes a Distinction without Difference:
This fallacy consists in attempting to defend... a point of view as different from some other one... by means of a very careful distinction of language... [which] is no different in substance.
There is NO substantial distinction between using free speech to say pleasant things and free speech to say hostile things precisely because the nature of free speech is that it is designed specifically to protect the speaker who uses speech to attack. And the reasoning is straightforward: It is better that uncomfortable facts be spoken than it is to allow those with power to suppress speech in the interest of free and honest discourse. Thus under US law, free speech is interpreted as expansively as possible including the concept of symbolic speech in the interests of truth, justice, and the American way. This is a mechanism to prevent the powerful from abusing the weak in society.