The nut broke off and had a small red ring/gasket between these connectors. I'm an noob but can follow instructions. Please give suggestions. Thanks
Below there is a water valve 
which may be used as an alternative if the above (cupper?) pipe is inconvenient to use.
You should not use a hacksaw to cut copper pipe for a sharkbite fitting, because supposedly the main thing that causes those to fail is the rubber o ring being damaged from rough edges. It's worth paying the $20 for a pipe cutter to not have to worry about a leak.