Sounds like a little one that misses their Mommy. My kids still used sippy cups at that age and not juice boxes to be honest. I worried about straws and poking eyes especially if you run a daycare. AND, when visiting my sister's house whose kids are slightly older, no joke, my 9 month old who was crawling got part of a juice box wrapper in his mouth and we didn't realize it. He cried for about 4 hours straight hysterically and so we went to the ER, pulled plastic out of the back of his throat. My sister's as neat as a pin but kids drop things and those little plastic wrappers are prone to that. I'd be careful if you have several kids and are offering juice boxes to kiddos or maybe you baby sit at her house and this isn't an issue. Anyway, it's so hard because different families have their ideas of independence. I put my kids shoes on forever because it was FAST. ha. Worked for me. They are older now and I assure you that they learned to put their own shoes on . . . so no harm was done in 'over parenting' a bit during those toddler years. But then I was with them all day so no one was inconvenienced by 'helping' with things. :>) So, I don't know what to tell you. I'd LOVINGLY help her with things you'd like her to do. But in general, day care/baby sitting should not be a time for a child to feel like their is something wrong with them when they are behaving and living as they do under their parents care (their normal). This little one is not yet even three. She may need more time with mom still. Sometimes that isn't possible for the work life of the parent--- but then YOU have to decide if you want to handle this type of child or not. There isn't anything wrong (from what I'm reading) with her. Kids develop a big differently. And she sounds like she feels secure for some reason and misses mommy.
This doesn't sound terribly out of the ordinary to me. good luck
Does she only have a mom or is there also a dad in the picture? What do you know about any assessments she has had done to address the things like not being able to drink out of a juice box? Do you think there is any chance she is being harmed by an adult in her life?