If you need to take time off for an emergency, just tell your boss it's a family emergency. Don't tell them what the emergency is unless you are required to. Say “family emergency” or something similarly awkward and say it's personal and you don't want to get into it. Don't lie, just don't say something you don't need to that can be used against you. “oh your sister is sick? she'll probably get better in a bit so do you really need this time off?” Never give more information than you need to.
If you need to take time off for an emergency, just tell your boss it's a family emergency. Don't tell them what the emergency is unless you are required to. Say “family emergency” or something similarly awkward and say it's personal and you don't want to get into it. Don't lie, just don't say something you don't need to that can be used against you. “oh your sister is sick? she'll probably get better in a bit so do you really need this time off?”
Never give more information than you need to.