In the next months we plan/need to refresh our https certificate and update our server OS version.
We are thinking about switching to Let's Encrypt.
It would be inacceptable, if clients could not access our https-API. Clients also include for example other webservers running on Java7.
I don't know, if upgrading the server OS to a more uptodate one would make any difference (ciphersuites etc.?). Planned proxy might be nginx, but this is not fix yet.
Is it possible/what to watch out for/best practicies/recommondations/alternatives?
Edit: To make it more clear: I can set up test servers and for example try accessing it in my webbrowser but I don't have access to all clients accessing the website because those clients are run/developed by other companies. I worry about wether clients like Firefox/Java7 webserver/Internet Explorer/etc. will have problems accessing the website. Clients are written by the companies using them.
Edit2: From the nature of our site, we can assume that all client-software access the website at least once per week (normally each day) and we can install monitoring software like wireshark on the server if nescessary if it helps.