Can someone help me with this? I recently heard a clunking noise in the rear tire and I found this...I've looked at OEM honda parts but couldn't find anything that these rubber thingy should be here....and why is it different on both sides?

 

Those almost look like spring spacers that have come out of their proper position. If so they're not stock. People put them into the spaces between springs to lift the car up a bit.

 

Would it hurt anything if remove them? How do I put them back in place...remove tires and squeeze them back in place? It is that simple..which one is right?

 

I'd remove them & see how the car rides. If something is wrong with the springs, installing some kinda spacer isn't the right way to fix the problem.

Like you, I can't find a legitimate part that looks like that.