No. The red wire from your new tail light is strictly for the brake light. That wire connects only to the Jeep's brown-red brake light wire. It lights the tail lights whenever the parking lights or headlights are on.
The new turn signal's black tail light wire connects to the Jeep's black-yellow tail light wire.
The TJ controls the rear turn signal by flashing the appropriate side's stop light, there is no separate turn signal wire in the Jeep. I dunno how you'll get that fancy Flow Turn Signal function to do its thing.
