ook here for a moment.
http://c.mhcdn.net/store/manga/8/60-604.0/compressed/nnaruto-3620083.jpg
You can see bubbles around Rin, sort of. I mean you can see the mist and steam stuff everywhere, but look specifically for bubble-shapes, you see them only around Rin.
now look here on 605
http://c.mhcdn.net/store/manga/8/60-605.0/compressed/r007.jpg
He says "just when we finally managed to get our hands on it"
"it"
Bubbles...tailed beast cloak?
IT=tailed beast?
We know now that killing a tailed beast host doesn't permanently kill the tailed beast, it only delays it from going free.
Rin might be the 3 tails Jinchuuriki before Yagura. Remember, Yagura was under Obito's control, Obito may have extracted the beast from Rin's corpse(if that's possible), and put it in Yagura.
Kakashi thought, for an instant, that the only way to stop the mist from grabbing Rin and getting a tailed beast is to kill her.
They likely want her body because they might be able to grab some residual chakra of the tailed beast, or might be able to discern the location where it will appear by tracing chakra.