This salesman was right this time.
Other than the freewheel, you may need a new chain because your existing 9-speed cassette may not have a large cog as the new freewheel does. You need to compare the 2 largest cogs. If you need a new chain, you may need a longer cage derailleur to take up the extra chain. Lastly, if you are good at tuning the derailleur, you may be able to keep the STI shifter because it is 9-speed. The Suntour freewheel can be 7 or 8-speed only. I bet you need to use a Bar End shifter and use the friction mode only or get an 8-speed STI shifter. The newer BE's index mode will not work with the 7 or 8-speed cog set unless it is one of the old old Bar End shifters. Don't forget the correct shift cable and housing.
It is a bit tricky to attach the Bionx brake sensor on the STI so be patient. Watch all the installation videos on Youtube may help.