I can power off my 1070 using the remote with no issues but the same button does not power on the unit.

To power on my unit, I can hit the "AM/FM" button, which turns on the unit and switches it's input to radio. I then have to hit the "DVD" button to go back to my desired input.

Anyone have this issue before, and is there a fix?

It's a bit unorthodox but as designed. You should be able to use any source selection button including "DVD" to turn on the unit though.

From the Outlaw 1070 manual:

As John points out, any input button will turn the unit on while the power button will turn it off. These discrete power on and power off commands are provided to assist with writing macros for universal remotes that support that.