So, if you start a new job on July 6, for example, you actually would not be eligible for insurance until September 1. Most companies like to start the new insurance at the beginning of a month, versus mid-month.
Just so you know, insurance companies (carriers) recommend a waiting period to employers because enrolling new participants takes a bit of time to do and costs money in administration. So, a waiting period ensures that the new employee has time to select what coverage she can afford and wants to have and that the new employee lasts in the job for at least the waiting period.
First-day coverage is just about unheard of these days. Many companies actually have a 60 or 90 day waiting period for benefits--so this sounds like a good company/employer.
Hope that the new job works out well for you in every respect.