For anyone who may come across this, I've found the answer to the problem and am sharing in hopes that it will save someone a lot of time and hair loss.
In my case, the username I placed into TestRail integration windows was not working. The reason for this is because I was the first user in Jira --> admin.
If I go into the list of users in Jira I see my username as "abrindell" and I use "abrindell" to login with successfully. This is actually NOT what I need to use for integration from TestRail. If you are the admin user of Jira, go into your profile settings and you'll see that Jira has a different username for you --> admin.
Once I placed the value admin into the username field of the TestRail integration settings PUSH defect worked like a charm.
I hope this helps someone.