It looks like egg membrane from a shell-less egg, and other matter. I would make sure that the hen is eating layer feed and getting some crushed oyster shell on the side to make sure her egg shells are strong. Soft or shell-less eggs can be common close to molting and if there has been any infectious bronchitis in the flock. It also can occur if there is a shell gland problem in the oviduct. I would try the calcium. If I were going to use an antibiotic, I might check into using Baytril. Here is a thread about it, where to get it, and dosages:
https://www.backyardchickens.com/threads/baytril-enrofloxacin-sources.959119/
https://www.backyardchickens.com/threads/baytril-enrofloxacin-sources.959119/
