She will be fine! The first day is always emotional, it's a shock to the system. She is not the first, nor will she be the last, to fight back tears. I remember a classmate on our R-Day that could not stop crying and I honestly wasn't sure if that person would make it through week one, let alone swab summer. Fast forward seven years and that classmate is a successful Coast Guard officer. Write your daughter letters, encourage her that she can do anything, and remind her that the clock cannot be stopped. Soon enough, swab summer will just be a blur of memories.
As a side note, it's moments like what you've described that make me question the "10 min goodbye" after the swearing in ceremony. A swab shows up early, gets whisked through station after station by people yelling at them, using nautical terms they've never heard of, and calling them out on every mistake. Then, as soon as you start to adapt and somewhat internalize the experience, the swab is thrown back to their parents for a brief hug and a kind word. Ten minutes later, they're thrown back into the new world of time demands and stressful situations. It messes with your head, or at least it messed with mine a little bit. I get that it might reassure parents but it can also terrify them if their kid is a mess. I had a swab come to me right after his swearing in ceremony saying he wanted to leave and that he was going to go home with his parents that day. I told him that was not an option, he was officially a member of the military now, and that it would be at least two weeks before he could be out-processed. The look on his face, you could tell he hadn't really understood the gravity of swearing in and taking the oath.