Is there a gaming pc for 300 pound max that will run minecraft / day z on normal settings?

Tons. Minecraft isn't intensive, and Day Z is mild. Most computers in that range are capable. Just be sure to get windows and not linux or chromebooks.

I don't think so.

Yeah pal! I don't know about pre-built like. Mine craft isn't graphically intensive at all, my 5 year old laptop can run it, it runs delightfully on my £350 gaming pc. As for DayZ the FPS can drop on many pc's as it's not very well optimised, but you can probably still run it at a sturdy 30-40 FPS. There's a £300 build on YouTube you may want to check out, it's by JackFrags and it's called something along the lines of "build your own steambox".

If you look at an older computer with older specs but a semi decent card yes. Check eBay or Amazon