I don't think there's any current, popular linux distribution out there that's incapable of running PHP5 and MySQL4.4+ (someone feel free to correct me if I'm wrong)... now whether it's included in the distro by default or not is another matter. That said, if you're going to maintain your own server, you should, at the very least, learn how to install and upgrade the software on it.
Personally, I've installed osTicket on Window, Mac, and a few different flavors of linux. some running apache, some running nginx.... my personal favorites being CentOS and Ubuntu.
For the absolute linux beginner, the aptitude package manager in Debian-based systems (which includes Ubuntu) is probably easier to learn and use.
Everything you need to run osTicket, includeing SSL and all the other optional bells and whistles can be installed via aptitude.