I can tell you a lot of people have had issue with aftermarket MPEMs, The start button is an easy test but the MPEM needs professional gear to check. Do you have a local garage that can check out the MPEM, not really sure they can check an aftermarket one either? Do you know of someone you can swap out the MPEM with your year and model, that has an OEM or a working MPEM one at least? You could also have your old one checked out unless you already know for sure its bad.
Do you have a multimeter? You can test the start switch buy using the continuity function, it is the one that giving you a tone when you connect the meter leads together. Find that and connect one lead to one of the wires and the other to the other, then press the start button, you should get a tone. The switches are normally open, if you get a tone when you connect the leads without pressing the button, then switch is bad, meaning it is closed. The test in the book (the maintenance manuals are free online) uses a resistance check but I find this easier.
You can also check the solenoid with a multimeter, get the book because the resistance values can be different by year and model. A quick test is to check your old one against your new one to see if there is a difference, more than a coupled ohms off. The books shows you a really easy test for the solenoid.
While your at it, do clean and verify good grounds, they create ghosts all the time.
