Do my landlords have the right to tell me how often my boyfriend can spend the night?

If the Landlord is your Mom or Dad, I would say yes. If not, they should maybe mind their own business.
If it is every night, they might think it is 2 tennants.
Maybe have him sign onto lease?
Landlord might want more money for Electric, Water,.

Well actually yes, many rental agreements restrict the number of days a guest can be overnight before they are considered a roommate. If you are renting based on one person, so why not see about adding them to the lease or rental agreement, pay what ever additional they may want.