My insurance renews on July 1st as well. My first baby was born in the third week of July, and yes, I had a whole new deductible to meet at that point. The nice thing in my situation is that I don't pay co-pays for my office visits, and my office doesn't bill for any of the visits until you have the baby. So the large bill from the OB office actually went toward the deductible that started after July 1st, even though the majority of my visits happened in the previous plan year, before July 1st.
The lame thing is that I had a hospital visit and a couple ultrasounds before July 1st, so my deductible was close to met, and I had to start over when she was born.
Not sure what date they would use if you went to the hospital on June 30th but didn't actually have the baby until July 1st though. I know that as far as the stay goes, it didn't technically start until she was out. The standard hospital stay is two nights. If I had her before midnight, that would have been the first night. I got to the hospital around 5 PM, but didn't actually have her until around 1 AM. So that's when my official "stay" started, and I stayed two nights after that. Based on that, I assume the stay was billed for the day of birth, and not the day of arrival. I don't have the bills in front of me to check that though ;)
What might get tricky is if you have an epidural or any other sort of procedure on the 30th, but don't have the baby until the 1st. I *think* that it would all be billed as part of your stay, and that would probably be after the 1st, but I'm not sure, and all insurance is different anyway.