Hi - I have a 2011 GSR. It used to be my daily until 2020 when I got a Tesla Model 3 Performance and I let my husband start using it. Well, I’m divorcing him and haven’t been great about taking the evo out since he moved out. The Tesla is in the shop for a few weeks and in need to drive the Evo to work.

It wouldn’t start, so I replaced the battery. Now the electronics are all working, but it won’t turn over. It makes a high pitch squeak and the dash goes black when I crank it. I replaced the relays and checked all the fuses. I searched the forums, but no one with a similar issue seems to come back and say how they fixed it.

I was hoping this would be a quick fix. Any ideas from the video?

 

The squeak isn't necessarily concerning depending on where it's coming from, it could just be your fuel pump/fuel system priming and trying to start.
Since you're not getting anything in terms of the vehicle turning over, I think it's safe to assume there's something wrong with the electical relating to the starter.
If it were me, I'd try jump starting it regardless of the new battery, if that doesn't work, triple check all fuses and maybe in the worst case replace the starter to troubleshoot?

 

That looks like a bad connection somewhere near the battery/fuse terminal or some relocation-mods. i've seen bad electrical/starter situations on my car but never seen lights go out like that.
maybe a mobile mechanic or a friend locally can help you. sometimes the problem area is easy to find but kinda hard to make guesses online