Has Catalina broken the logging into external machines?
No, Catalina has not broken the logging into external machines in general. The problem must be local to your machine or your network. I regularly use the commands you mention (ssh and mysql) to connect to external machines from a Mac running Catalina - so in general it works just fine.
I would start by ensuring that you have:
a) a working internet connection (you can browse web sites from the Mac)
b) not running Little Snitch or similar software that blocks outgoing connections
c) not running a firewall or similar on your router or network that blocks your external connections
Then debug by examining exactly what doesn't work:
a) can you ping the IP address of the server you want to connect to?
b) can you ping the host name of the server you want to connect to?
c) can you connect to other ports on the server (for example a web site)?
d) does the server log tell you anything when you try connect?
e) if everything fails, use tcpdump on the Mac and on the server to see network packets going back and forth
From there it should be possible to figure out what the problem with your setup is.