We are building our own home to earn equity program. My husband hung our drywall using 1/2 inch sheets on the ceilings with 24" in between joists. We have been told by our drywall finisher that the ceiling will sag because we were supposed to use 5/8. Is there anything we can do now to correct this without redoing?

In my opinion, you are just Fine. As long as you've done the proper amount of 'edge and field' nailing (or much better would be using screws) there should not be a problem. Without knowing more about the reasons for this persons statement, I can't really say much more. Except that I've used 1/2" (Type X - fire code) drywall in a number of projects and it's Always worked just fine!
And as for 'finishing', you could probably do that yourself too - and save some money. But that's for another posting.

It is already up there so just leave it. If it sags then you can replace it with heavier wall, 1/2 should be OK.

Sometimes "Nailers" are Nailed to Joist, Perpendicular to joist. Then Drywall.
Not sure, If it does sag, and you want to fix. May be easier to secure nailers over existing
Drywall, and another layer of drywall.
Possibly install Nailers as decorative over drywall and paint. Forget second layer.

A lot Of New Homes We Run Into Today Are Built Using Trusses And Are Also Sheetrocked With 1/2, You Really Should Have Nothing To Worry About,
5/8 Is Used For Commercial Work, Also To Use As A Firewall, Or Help With Soundproofing,
Your Fine No Worrys

I am drywalling my bathroom and installing 1/2 moisture greenboard on the walls.On the ceiling should I use 3/8 or 1/2 inch moisture board.I also used a faced insulation on the walls should I also use plastic as a barrier.

Your drywall was correct in saying your ceiling will sag or pillow down unless you used no sag 1/2 rock as the other folks stated you are fine are they drywallers do they know 5/8 has fibers in it so it is stronger and tends not to sag

dwingconcrete
Use 1/2 green board on ceiling. The facing on the insulation is your moisture barrier.