I have fliiped the breaker, and I still get no power to the plug inside the AC/Furnace closet. I have run a small extension cord to the nearest plug so we have heat for now. I changed the plug inside the closet to a new one from home depot, but I still have no power. Don't know what to do from here, any help would be appreaciated.

Check the electric circuit from the house electric panel to the plug. Make sure the breaker is turned on so the voltage will be on to the plug so the unit will run. If that fails call an electrician. Extension cords are BAD news.