This question does not have an answer because it's a matter of opinion. There is no good way to measure how free-speech or anti-free-speech a president is. Here's what I can give you:
The organization 'Varieties of Democracy' does research to calculate a 'freedom of expression' score for many countries. They have data on the USA's freedom of expression from 1789-2021 currently available. The Biden Administration averages a score of '.92', which is larger than the Trump administration's average of '.9'. However, both Trump and Biden have the lowest scores since the Nixon Administration.
https://ourworldindata.org/grapher/freedom-of-expression?tab=chart&country=~USA
In terms of why John Adams is mentioned, he does have the lowest freedom score in US history. He passed 'The Alien and Sedation Acts' which hindered a lot of freedom in the United States. These acts limited immigration; but, more importantly, in terms of free speech, it made it illegal to criticize the government. You can find these acts here: https://www.archives.gov/milestone-documents/alien-and-sedition-acts
This is a long way to say this:
1. There is no true way of determining if this is true or false as there is no correct way to measure how 'free-speech' someone is.
2. Using the 'freedom of expression' score during each president's time in office - no, this claim is false. One could argue, though, that Biden is the second most anti-free speech president since Nixon.