Yes, the shell will come out on its own.
Correct, it is a tablet. You just pop the tablet in her mouth and she'll swallow it!
That's exactly what you want!
What I would do is get some calcium citrate +D3 and give her that once a day, even after that egg shell is out continue to give it for a week or so after.
You can also try to lubricate around her vent to help her get that egg shell out.
Do not pull it out, as that may do more harm than good...