Eaton won’t go into reverse after breaking reverse light switch

Some switches have a spring and a ball bearing swaged into the end.
Look at piece that you pulled out w/easy out.
Is there anything in it?
Sure hope you didn't loose a small spring and a ball bearing in trans.
Do you have the new replacement part?
Look on the end and see if there is a ball that would press in and activate the electrical switch.
You may have to get a strong magnet and go fishing it the oil.
Go in between where trans would move - to go into reverse.
Jiggle the slider to free up anything stuck, with magnet close by.

 

Like @rolls canardly said. Sounds like the ball and spring popped lose and lodged somewhere on the rail in there.

I like the paper clip and coins that fell through the shifter boot slowly being fossilized in the oil. Lol!

 

Unfortunately it sounds like the detent/spring from the switch is jammed up in the rail. If you can't get them out somehow (magnet, pick etc) you'll have to pull the shift bar assembly.

To make matters worse, the transmission might actually have to be dropped as there might not be enough room to get the cover out.

 

That was the obvious answer, but not the only/or correct answer.
You took good pix; looks like U took top cover off.
Drain the oil w/plug on bottom.
Get a can or 2 of brake kleen, and wash down all the slime,
put the 6" straw snozzler on can, and pay particular attention to areas
where gear contacts gear. move assy. with a big screwdriver to see
areas you cannot see. Don't spray the oil off any visible bearings.
Refer to your "before" pix, to get everything back in position,
and jiggle everything around, to get the lid back on it. Don't force it.
After you find your issue, fill with fresh oil.
Spray sealant on gasket before U drop top back on.

 

So.... the threaded portion of the switch is still in the housing?
And you were able to recover the inside parts? Good.
Spray it with spray oil so it goes in around the threads.
Maybe some will go in, but it's probably tapered pipe thread.

You will need a set of easy outs, so there are a range of diameters.
When you put your brand new easy out, (that you are going to buy,)
so it has sharp new cutting edges, insert it into this broken piece
so that it rests about halfway up the spiral. Don't turn it.
Take a hammer and drive it into the metal like a nail; that hard.
If there are obstructions, take a bar, set it on end, as close to centered as you can, and hit the other end of bar.
Don't get crazy and break it off in there. Firm, not crazy hard hits.
Get the easy-out seated in the threaded piece.
Object is to get it in - so it can't turn without removing threaded item.
Take a large tap handle or 12" adjustable and try to turn it.

 

Re- read what you said;
Yes; try to pry it - to confirm if it is threaded.
If it is, follow above.