Can I claim my live in boyfriend as a dependent if he was in jail for 7months out of the year and I had to pay all the bills

No, you cannot claim him as a dependent since he did not live in your home for the entire year.

But I thought that since our home was still considered his residence, where he received mail, that would be ok? Like if someone had to leave the residence for some other reason, but still used their home address as their residence.

Being incarcerated is not considered a temporary absence from the home.

Additionally you would have needed to provide over one-half of his support for the year.  You did not do that since the state/county/city department of corrections provided his support for 7 months.