Hi Kate,
Thank you, yes I did see that thread, and led me to verify the Internet Options for TLS and SSL (they were already previously enabled/disabled respectively).
I also found the following thread https://community.logmein.com/t5/Pro-Discussions/LogmeIn-Client-shows-offline-Cannot-connect-to-it/td-p/308877, which shows the same error in the UI, however the solution, if there was one, was not shared.
There was also this thread: https://community.logmein.com/t5/Pro-Discussions/LMI-Control-Panel-can-t-connect-to-network/m-p/309411, which also shows the same issue, along with a Win 7 machine.
In response to your question, the other systems are not Windows 7, the issue seems to only be affecting Windows 7.
It's interesting to note that the issues above, including the one you posted, are all within the last two months.
Thank you for your attention to this.
*edit, just after posting this, I noticed another new thread by user philbud with apparently all of a sudden new connection issues... also Win 7 machine.