She's 14 -- too old for you to micro-manage her wardrobe. Yes, she's making a mistake, but she's old enough to make it for herself. Cheerfully tell her these rules:
1. She may choose her own footwear for the trip.
2. This trip is going to involve heavy walking, and flip-flops are likely to give her blisters. If she develops blisters (or other achey-foot problems), she is NOT to complain or diminish anyone else's enjoyment in any way.
3. She must bring along the comfortable shoes. She doesn't have to wear them, but they must be in her suitcase available, should she decide to wear them.