mnsparky
If the branch circuit feeding the 220 outlet is a dedicated one you may be able to take one of the wires and turn it into a neutral by taping it white and placing the neutral bar in your panel also making sure to put that re-identified wire on the neutral side of your new outlet. Then take out the 2-pole breaker that was feeding it and replace it with the right sized single pole breaker and do the same with the outlet in the wall. Make sure the wires are sized correctly for the amperage of the new unit.

Coolness
You can buy 220 volt units from HVAC wholesalers. They run more efficiently and draw less current from your electrical service. But if you don't want to do that, then read on: Change the 2 pole breaker to 2 single pole breakers. Take one of the wires (preferably black) and hook it to the single pole breaker and the other will (could be red or white) go to the neutral bar. Leave the ground alone. Now change your outlet and make sure that the wire that went to the neutral bar is connected to the silver terminal on the new receptacle. The wire that is connected to the breaker will go the the gold (hot) terminal. Take the old ground wire and ground it to the new receptacle. Mark the neutral wire with white tape if you had to use a red wire. One consideration. 220 volt units draw less current than the 120 volt unit of the same capacity. Make sure the wire is large enough to handle the current draw of the lower voltage unit. If you use a 20 amp breaker then you have to use 12awg wire. If the old unit had 15 amp breaker then it might have a 14awg wire. Be safe and do it right.

John himself
Both answers you got would be violations of the national electric code. And check the ampacity of the existing wires against the load of the new AC.

plain jane
Know when you're in over your head...get a LICENSED electrician to do the job. Check with your homeowner's insurance agent...I'm 100% sure that he/she will tell you that should there be a fire or shock incident, YOU will be 'on the hook.'