Screen Replaced - Not Charging or powering up now!

I recently sent my son's 5C to a repair shop and when we got it back the phone doesn't charge up or go into DFU mode. Normally I would've done the work myself, but I figured having a few spare minutes with my kids was worth the time I saved vice doing it myself.

The phone wasn't charged when they gave it to me to test, so I made the cashier charge it to 5% so I could at least make sure the phone worked. Assuming that everything was ok, I drove home and placed it on a charger.. it sat their for 5 days and never charged!

Thinking it was the charger (5C's are notoriously fickle with non-supported chargers) I bought an real Apple charging plug and cable and it didn't change a thing.

My next step was to put the device into DFU mode, but iTunes doesn't even recognize the device.

I decided to take the phone back to the shop that "repaired" it and got all kinds of excuses why it didn't work (bad charging port - they replaced and still didn't work, bad home button - yup once again replaced and still didn't work, and finally new battery - which still didn't solve anything).

They had the phone for 2 weeks and all I have to show for it is being $100 poorer and a bricked phone.

Suggestions?