If your hours are at the discretion of your employer, it's probably not uncommon for you to be let go if you can't meet their expectations. Just because others have discussed availability and are exempt (for whatever reason) doesn't mean that you'll necessarily get the same treatment.
Example: Where I work, we can do compressed workweeks (work 8.5 hrs a day instead of 7.5, and get every 10th day off in lieu). A lot of people ask to do them, but if there's already someone who wants, say, every other Friday off, it's unlikely that anyone will be permitted to take the same set of days off. Whoever got there first gets the day, and if you're the second person who asks... too bad.
Sucks for you, as you're just trying to do the right thing for your child, but... there's probably nothing wrong with it, unless you have a contract that says that you have a hand in deciding your work hours.