Does that happen any cell or just one with a specific command?

Another question, does Mathematica kernel work on both machines independently?

It sounds like you've diagnosed the source of the issue quite well! We know the general source is the problem machine, and we also know the problem isn't with the commands being given, as it happens with any cell, regardless of its content. Additionally, since it works on the Mac without issue, we can rule out problems with the Mathematica software itself.

Have you tried checking the system logs on the remote machine for any error messages or warnings that might give more insight into why the Math Kernel process is quitting? It might also be helpful to ensure that the problem machine is set up properly to handle the remote connection. If the issue persists, reaching out to Wolfram Support could provide further assistance.