You can put that white rubbery thing on the list of freaky, disgusting things that can come out of a hen's butt. If you dissect it once the hen completely expels it, you should find it's more of an elongated, collapsed egg membrane with perhaps a little albumen inside. All it really is is an attempt at an egg and near miss, a one-off thing that means little in the larger scheme of things.
When I see something like that, especially if the hen is acting as if she's in some discomfort (hanging out in the nest or in a corner not being active), I give her a whole calcium tablet, your choice of citrate, carbonate or gluconate, to help strengthen her contractions so she can more easily expel the last of the strange stuff. If it stretches into a second or third day, and it could if some of the material has lodged inside her, continue the one-a-day calcium until she begins acting herself again.
