I don't know where you live or what the laws are in your state, nor whether you belong to a union. If you're in a "right to work" state, you can be dismissed just like that. If you are represented by a union, you have a collective bargaining agreement which specifies the steps of discipline. I'm a union steward in a hospital in a "union" state, so as a public employee (which I am) I would be subjected to progressive discipline even if I weren't protected by a contract. This means that there would be verbal warning, followed by written warning, followed by a pay reduction for a 3 month period, and then pre-dismissal. However, if I were new (within my trial service period of six months or a year) none of this would apply. Similarly, if I did something really egregious, like came to work drunk and accidentally killed a patient, I could be dismissed on the spot and subject to criminal charges.
Okay so, all that being said, I have to assume the period you were out of work was not a FMLA-covered absence. in order to qualify for FMLA you have to have worked at the facility for a certain length of time, and it depends upon hours. Generally six months isn't long enough.
The details you've given are a little sketchy. What I would say is that you are being given a second chance but unless you have union representation, you may correctly interpret this verbal warning at face value. On the other hand, it is unrealistic to expect ANY employee to never be sick, or never have to care for a sick family member for the duration of their employment. So you need to talk to your administration and clarify the terms. If they want to place you on a probationary status, that's reasonable. But you have to pose the question, what if I am too sick to work? Do I even need to bother to call in sick? Is there a timeframe on this? Research your state family medical leave laws in addition to the federal law. Educate yourself about what is legal and what is not. If you find yourself terminated while in a family medical leave status, you have grounds to sue for discrimination. On the other hand, unless you get it in writing what the expectations are (I would insist) you will have nothing to show an attorney or your union or the EEOC if your employer decides to cut you loose down the road.