Linolium and vinyl will break down over time, and it can be a relatively short time. As it breaks down, moisture WILL get under it and can do damage to a wooden floor. Paint is OK, but it really doesn't seal the gap where the walls meet the floor. I would not use caulk, because as it ages, it starts to come loose, and as was mentioned, the chickens WILL notice and try to eat it. What I use, and IMO, it is THE best cover and protector of a chicken coop floor, Is a rubberized roof coat product I got from LOWES (Blackjack #57). This is the stuff used to seal roofs. If it can stand up to 4 seasons of all kinds of weather on a roof, from super cold to blazing heat, protecting a chicken coop floor is nothing to this stuff. It totally seals the floor gaps, and the gap where the walls meet the floor with a coat of rubber. The gap where the walls meet the floor is where insect pests like to hang out. They can't do it when the gap is sealed shut. And it is NOT slippery.
I have had this product on my coop's floor for over 2yrs, and it looks as good as the day I put it down, although I only get a look at it twice a year when I clean out and replace all the bedding/shavings. To apply, I just dumped a big dollop on the floor and pushed it around with a roller. I applied a heavy coat to the whole floor. If I wanted to, I could fill my coop with an inch or so of water. You want the BEST floor cover/protector, This is the way to go.
Jack