What happens if you don't pay your civil liabilities? Civil vs criminal is a silly distinction when it comes to discussing right suppression. Jail time is not the only way to suppress a right.

Civil and criminal law are separate things, absolutely. But the first ammendment applies to both civil and criminal law - it is a limitation on the government's ability to create laws, not just a limitation on the government's ability to pursue criminal penalties.

Alex Jones is only liable because there exists a law that the government created that says that defamation is illegal. Since this is a law, it could have been in conflict with the first ammendment - and, in fact, there have been legal challenges on this very line that reached the SC. But the Supreme Court has found that this is an acceptable limitation on the first amendment rights, with the specific limitations.

But, for example, if the US government wanted to adopt the English law on defamation, it would not be constitutional in the USA, it would run foul of the first ammendment.