While I am watching recorded programs on my iPad about 20-35 minuted into watching the program the wired network connection on my iMac crashes and can no longer access local network or internet. I have to reset my computer to restore network connections.
When you say "crashes" and "reset" what do you actually mean--what actually crashes and how are you resetting your computer? Do you mean your Mac just drops its network connection and disappears from your home network, i.e. other Macs or iOS devices on the network can't screen share in anymore? Is a reset a restart? Is your crash a complete freeze of your entire system, a kernel panic, requiring a force restart, or can you still move your mouse cursor around and are other apps still up and running just fine?
It's been a while since I've used the EyeTV iOS app for anything, in fact I don't think I even put it on my iPad 3 once I traded up to it from the 2, but back when I did I never had a problem with live TV streaming to the iPad 2 from a 2011 Mini in either 10.6 or 10.7, wired to an Airport Extreme like your iMac. Before there was an EyeTV iPad app I streamed my recordings to the iPad with Air Video, that's the first thing I'd suggest you try--take EyeTV out of the equation, stream your recordings with a different app like Air Video or Plex, and see if you get the same behavior. Also, back then El Gato made you specifically "prepare" any recording you wanted to stream to an iPad in advance
which I thought was a waste, so to this day I just kept streaming my EyeTV recordings with the Air Video app, it's always worked perfectly transcoding those recordings live.
Make sure you uncheck sleep, putting hard disks to sleep and screen saver while you're troubleshooting...
How long have you had your 16GB of RAM in the machine?