I have a 4-year old and an infant. In our house, juice is a special treat, for a sore throat or a special meal. Even then, the juice is cut with 50-75% water. We just never have juice in the house. Milk or water are the only options (and at restaurants I say, do you want milk or water - I never say juice is a choice).
I suggest telling your daughter that you have a new policy, similar to this. Explain to her why you want her to cut back on juice and increase milk/water (health) and ask her what would help (what does she think she should do?). Show her how much sugar she is drinking - let her measure it into a bowl. You could introduce gradually and tell her okay, today you are going to have two juices, then the next day one, etc. Or buy low sugar juice boxes for the week (like a 6 pack) amd tell her that she can have them whenever she wants - but when they are gone, they are gone, no more! So she can have six in one day, but she will be out of luck for the rest of the week.
Personally, I oould go cold turkey. She will drink when she is thirsty. She needs the milk. And if she is drinking all her liquid in juice, then that is very unhealthy in terms of sugars. In the meantime, offer her lots of water foods, like cucumbers, fresh fruit, low sodium soup, etc. I assume you already offer lots of dairy, etc. When she asks for juice, give her a comparable peice of fruit. Stay strong and let her know you are serious. Also, introduce her to smoothies. You can have fun making these together. Use whatever milk you like and fat free yogurt, fresh fruit, and ice (no juice) in a blender.
If you do decide to lay down the law, however, I wouldn't recommend taking her out in the heat if she refuses to drink. And this could be a bargaining chip - if you want to go the playground, you need to drink this much water before we go and then bring water with you.
As I said, I would draw a hard line on this, but you can still try to make it fun for her. Let her do a "taste test" for you and select her facvorite drink. Give her seltzer water in different flavors (she might think they are like juice because you can get berry and lemon flavors) and different types of milk (skim, 2%, soy, almond, etc.). Tell ehr she can drink whichever one she picks. Let her pick out some swirly straws and tell her they are only for milk/water. Give her a pretty pitcher and tell her to pour her own drinks.
PS - I just wanted to add that I clearly agree with the other moms who say this juice thing is ridiculous and the juice should be gone quick!! And by the way, chocolate milk is worse than juice because it is not even natural sugars - its fructose or corn syrup or cane sugar. However, I do think it is important to make this a good experience for her and not a punishment. It is not her fault and she doesn't know better than she has been exposed to. This is a hsbit, a comfort, and maybe somewhat of a physical addiction. To cut out all that suger will make her feel lousy for a few days. It HAS TO be done but it will be tough for her. So do it gradually and lovingly.
Good luck - I think it si great that you are trying to make a change for the better.