Removing Float Floor

This might sound like a bit of a dead-end question but I'll try anyway.

We are trying to take the floor out of our trailer prior to painting. It is a marine-ply floor with spray rubber on top.

We have discovered that it is screwed into box sections and the screws have been covered by extra spray rubber layers put on by the previous owner.

Anyone have an idea how we can get these screws out of the box without wrecking anything? I would really like to keep the floor in one piece and of course there is nothing wrong the with box so I don't want to hurt that either.

could you maybe just cover it with a painters tarp or cloth instead of removing it?

I'm not sure I understand what the area looks like. Would you be able to post a picture(s)? I will get my man to take a look and see what he sais. He's worked on all sorts of motor vehicles in the past so I'm sure I could give you some good advice.

Thanks for the ofer My2Geldings. Went to take a pic today, but husband has decided HE knows what to do. Must admit, he has over half of it done but now he's up to the front and come to a halt.

He had better not break that floor, or he is dead!! I swear!!