I recently switched from Windows Home Server to Xubuntu as my file server in a small home network. I got Samba configured easily for all users' shares using the Samba-config GUI (it's a mixed Linux/Windows network), and am generally happy with the Xubuntu box, with one notable exception: every few days, it informs me that updates are ready to install and after I click to install them, a reboot is required, necessitating a few minutes of server downtime.

Although it's just a home network, I'd rather not have to take my file server down. I realize I could probably have avoided this by using a non-graphical distro like Ubuntu Server, but although I can muddle my way around the command shell when necessary, I'd rather not do so and prefer a GUI.

So, I'm wondering: would switching to Centos for my file server avoid having to reboot every few days to install updates? Since it's based on RHEL, I imagine it's designed to avoid the need for reboots. I've always used Debian-based distros and am more familiar with them, which is why I chose Xubuntu in the first place, but I'd be willing to go with Centos if it avoids downtime.

Any serious distribution, Ubuntu or Centos, GUI or commandline-oriented, will present updates to you on a regular basis. You can opt for not installing the updates, at the risk of running into bugs and security problems. The less software you have on your system, the fewer updates are required, so that you may want to move to a non-graphical server. Whether it's Ubuntu or Centos is irrelevant.

That's kind of odd - most servers only require reboot for a new kernel... Shouldn't happen that often really...
You could try Centos as an alternative and see how that goes.
On RHEL/Centos you can see what updates have occurred

To expand upon this -- the reason you are being asked to reboot is because Xubuntu has installed an "update manager" on your machine which installs updates then prompts you to reboot -- this is probably because some of the libraries and executables it installs won't be reloaded until a reboot or, at least, an X11 restart has taken place so it mimics Windows and prompts you to reboot.
You have a few options, you can safely ignore the reboot prompt and reboot when you feel like it (though depending how the update manager works this may result in an inability to upgrade more or lots of annoying prompts), could use something else (either synaptic or apt-get) to install updates and reboot when you want or you could uninstall the update manager entirely and just manually update.
Personally, on any system I run (and mine are generally desktops) I uninstall the update manager and manually update using apt-get when it is convenient -- only rebooting when something I am bothered about is updated. So I rarely reboot my Pi (practically only for new kernels) and reboot my desktop and laptop when something is updated that I want to use right away as they are powered down fully every day anyhow.
Perhaps others will be able to be more specific about which updates require are reboot to take effect.