I don't remember them being open during the party we went to last year. It seemed like there were only a few quick service places open, mostly along Main Street. If they were open during the party, you would have to have a ticket for MNSSHP.
Some of the TS locations will accept ADRs during the MNSSHP/MVMCP hours. Most folks here on the Disboard generally agree that having a TS meal during the party is not a good use of the party time. If you have an ADR for 7:00 pm or later on a MNSSHP/MVMCP night, each guest will need to have a valid MNSSHP/MVMCP ticket to be allowed to dine.
You can make an ADR at BOG on a MNSSHP night from 4-7 PM, but not after that. If you don't have a party ticket, you can go to dinner there if your ADR is before 7 PM, but will be escorted from the park by a CM after you dine.