Hi Gregg,
I'm not why this is happening. May be these questions will help narrow it down.
What version of Pro are you using? (I'm at 1.1.1)
Where is the data (shapefile, file geodatabase) you are editing?
When you hover your cursor over a green vertex, does the icon change, as shown below?
Can you right click on the line and insert a vertex?
What hardware are you on?
One observation is that your green vertex is a solid square, and mine shown below appears with an opening in the middle.
For testing, what if you draw a new line, not connected, and then move its vertices?
Rob