I am looking at using a couple of WYSE thin clients for broadcasting feeds and I am wondering if it would be better to put Linux on them or try running Windows CE?
I think HP thin clients are a better way to go! Specifically the T5720. WYSE thin clients are harder to work with when it comes to backing them up, or re-imaging them. The HP T5720 has a built-in utility called Thinstate that allows you to back up and image, and/or restore it, from a thumb drive.
Thin clients normally do not run Windows CE. They run Windows XPe (embedded). It's a very small, stripped down version of XP. Some run Linux as well, typically some flavor of Debian.
The big snafu with thin clients is that they have no hard drive, and use a solid state flash drive instead. These are typically very small, and have little room for additional applications to be installed. They are originally meant to run applications over a network, from a server. The flash drive is also write protected, unless you disable the write protection. Normally this is done temporally to save configuration changes. Flash drives have a limited 'write' life to them, and over time, if written to, too often, they will fail.
You might be better off buying an older PC off Ebay, for about the same that you'd pay for a thin client.