74HC595 to 5v pin shuts down uno
when I attach pin 16 or pin 10 of the 74HC595 to the 5V pin on the Uno ... the entire arduino shuts off

This sort of thing usually happens to me when I have made a mistake in the wiring.
This is easy to do as the 595 needs a lot of connections in a small space and it is easy to make a mistake.
In your case it seems you have a wiring mistake that is pulling Vcc low and the ATmega328 IC etc can't run.

I would

Re-check the orientation of the 595
Re-check each wire paying particular attention to the ground wires, especially the one that pulls 595 OE to ground (it's pin 13 on the 595).
If necessary, pull everything off the breadboard and start again with things in a completely different position (turn the IC's round etc) to make me re-think everything.


For fun, I set this up on a Uno clone. I made some mistakes in the wires linking the 595 to the Uno but careful inspection found the problem. I found it easy to mix up the pin-number on the 595 with that on the Uno. Using the same wire colours mentioned in the guide may help.
Here's it working