I think the OP worded her post badly and it's brought out a group of people who have had good reason for keeping health issues private or between them and their manager, who are seeing their situation in the OP's and reacting to it.
However, it seems clear from the OP's later posts that this woman has form for doing this. She may well have a reason, physical or mental, that she doesn't want to share, but, if it's a pattern and there is a feeling by the team that this always happens at crucial points, then people are going to feel frustrated and resentful. However, it's your management that are letting you down OP. If there is a reason that they're not sharing with you, then they have to ensure the rest of the team is supported in her absence. In.a one off situation, everyone pulls together, but when it keeps happening and she never puts the extra hours in, you create a division and that is terrible for team morale.
I've had to manage out an employee who was just lazy. She routinely arrived late and left early, took long lunches and spent most of her workday chatting in the kitchen. We were project based and he pulled sickies when the pressure was on, and it became a real problem. We had to put her on performance review, and it got to the stage where she had to report in and out to me, and we had weekly reviews to show her what we were expecting and where she was falling behind. It was horrifically stressful for all involved, and she felt very unfairly treated. The thing was, she was just totally unsuited to the job and the pressured environment. She eventually quit before we had to sack her, and I'm sure she feels hard done by to this day.
Whilst it was all going on it created huge schism in the team, and a group of people who always had each other's backs and were a great team started falling out and bitching.
For all the genuine, private reasons that people are off, there are people who just don't pull their weight. It is an absolute drain on the team and everyone knows it.
OP, if this is the situation, you anbu.