I am very upfront with potential clients , I tell them I have appointments with a specialist every x # of weeks and that I will always open late those days.

If I have to go to the walk in clinic I either close or have my DH replace me.

I really do try to be professional and give them as much notice as possible but sometimes that's just not possible.

I found it really helps to tell potential clients that I may close if my children are too sick and need my attention, if I am really sick,ETC...I always ask if they are prepared for this and ask if they have family in the area that could help them out.
I have lost only 1 potential client due to this and the others who became clients never ever complained when I took time off,whether it be for appointments, illness or my 4-5 weeks of vacation per year.