No employer is required to provide paid sick leave. No law requires it unless youre part of a union and have a union contract. You can have unpaid leave but youre employer can fire you too, especially in states where employers can fire 'without cause' or reason.
Sick leave is given when one falls sick and the doctor sees that he/she is not in a position to work for some time not matter the days after employment. After employment confirmation, one can fall sick and that means they will have to rest no matter the duration from the date of employment.
No, they don't HAVE to give you paid sick leave, and actually if you're paid on an hourly basis, it's unusual for them to give it to you. If you're sick, make SURE you call in by the time you were supposed to show up that morning, and let them know what's happening. Some companies demand a doctor's statement (or a note from your mommy) if you're out more than a couple of days. I got fired from my first job because I did not call in when I was sick. Others had been out and no one knew why, so I did not think I needed to call either... Wrong.