For minors, they may need an ID if checking in at the ticket counter with an agent, in which case any photo ID (school, learner's permit, etc) will do. For TSA, they just need to state they are under 18 years old.

As mentioned above, you should also be able to get a gate pass from the AIRLINE to accompany your nieces to the gate. I'd wait at least until the plane pushes back and, if you really want to, even until the plane actually takes off. Assuming they're flying UA and you have United club access, I'm sure the agents there can tell you when the flight actually lifts off while you enjoy a drink.

Aside: reminds of when my sister and I were in jr high/high school and we'd visit my bachelor uncle in Denver every couple of summers. Our older cousins would also visit. We'd pile into his '77 Town Car and visit everywhere - Durango Silverton Railroad, Royal Gorge, Aspen (great place to visit in the summer!), Central City Opera, etc. I'm sure you'll show your nieces a good time!