How would you react to an E2 who "smart mouths" you in formation?

Write him up. Unless he learns now, his career will be very short.

Help this 2 class Mosquito understand that i politely request that he help himself to the ground and move the earth untill i gain 2 inches on my arms
I would also request this 2 class Mosquito find and understand Burger Kink and McDonalds oh the Air force is always looking for quilified Q and A personal

Well first off this is what you get when people join that don't really want to be there or understand that they are a small cog on the big gear and what happens when a cog on the gear breaks it starts to slip by him saying no to a direct order he is causing the gear to slip. Now if he were in a fire fight and refuses to call in a air strike we all die so he should go out back and swallow his sidearm before he gets the whole platoon killed when he can't handle the real pressure he will be put in. If he wanted to serve his country but not be a part of the bigger picture he should join the wemons march not the elite of the elite his actions will get himself and his brothers killed in real time take him out back and beat the shot out of him oorah... you cannot be a pussy and defend our country it does not work like that

Whine and tell me to choose someone else? - Strike one
Tell me to shove it? - Strike two.

This is where my authority as an NCO comes in; I'd call the next ranking person below me, delegate the responsibility of relaying the assignments to the rest of the formation as not to delay operations. I'd then have private shit-bird stand fast with another person while I sought out another NCO or PL to accompany me as a witness. in issuing corrective action. Calm assertiveness and not tolerating behavior, I would provide him one opportunity to explain the circumstances (much like some form of amnesty opportunity) as to why I should not initiate the UCMJ Process and recommend Article 115 action.

As a leader, it's vital that my guys trust me to look out for them and guide them; conversely, I need to be able to trust that my guys are going to do what they're told and communicate with me. Honesty and respect goes both ways.

Obviously private shit-bird would, at minimum, develop some vertigo from "Front, Back, Go". Sometimes, Joes just want to see what kind of rise/reaction that they can get out of you, it like a cry for attention.

Instead of physical punishment (i.e. a smoking) I might say, "Very well, as you wish...you will stand here at the Position of Attention and make sure that the asphalt stays in place...all....fucking...day. I'll have the CQ escort you to lunch and then you will resume your post until recall formation.
You will render a salute to all officers who pass by and will give them the appropriate greeting of the day.
If you leave, then I will recommend a Field Grade Article 115 for 2x counts of Insubordination and one count of violating your 1st General Order. If at any time during the day, you wish to rejoin the team, you will sound off in a voice that is loud enough for the CQ/Runner to hear you. He/She will bring you pen, paper and clipboard; you will submit a written request which will then be passed on to me. I will consult the Chain of Command (both ways) before returning my decision."

Then you let the other guys in the formation know that since private shit-bird doesn't want to play that the mission still stands and that we all get to pick up his slack. You kinda create the peer pressure/Lord of the Flies thing. If the rest of the group performs, then reward them afterwards; private shit-bird gets leave and pass requests denied.

Captains Mast

He goes straight to Captains Mast and then 120 in the brig

Dismiss thee rest of them and keep PVT Smart ass. Then gave a verbal direction on what he will do.

Back when I was in, I would have slapped him upside his head and told him either do the job or face the ucmj for disobaying an order

P.T. HIM TO DEATH... SUPERIOR OFFICER COMPRISED OF SITUATION.

Throat punch sounds good!