It's a whole new ballgame when you've already got one child. This is a two-part question--labor and delivery, and sleepover while you're in the hospital.
Is your child in a mother's day out program? If not, perhaps you've got time to find one where he can go for at least a few hours a day while you're in the hospital to free up your husband to be with you. But your husband will have to leave your side when it's time for pickup--the beginning of the many, many compromises families have to make when you go from one child to two (or more).
I wouldn't recommend expecting your son to spend the night at the hospital even if staff allowed it. It's important for him to keep to his regular schedule as much as possible. Having your husband at home for him is more important than having your husband at the hospital for you, where there are nurses and nurses aides to assist you.
I had both of my kids at St. David's (central, on 32nd), and that's pretty much the routine we did when our second was born. I actually relished the couple of days in the hospital without my husband or older son, when I could focus on the newborn all by myself and know that my husband was doing the same with our other child.