You shouldn't have any problem swapping freewheels with
that RD-6207 (BR-6207 is the brake caliper), especially one
Shimano freewheel for another. The derailleur will handle
a 28T cog OK.
Just a followup. I may have missed this subtle point. Turns out (my fault for being careless) my freewheel on the bike is a 11-21 Suntour. I ordered a 12-28 Shimano. The width of the new freewheel is such that the smallest gear rubs on the chain stay. In other words, the axel is not the first thing the chain stay hits, it is the smallest gear, which makes peddling difficult
Does this sound accurate to everyone? I suppose i need to find a suntour freewheel?
By any chance is there a spacer on the hub - like a thin washer - under the freewheel? If so try removing it and see if it still rubs. You don't say how much too wide the freewheel is, but I assume it's not much. You could mess around with axle spacers etc. and get it to work but you'd probably also need to re-dish the wheel slightly. Probably easier to find a different freewheel.
You might also post this in the mechanics forum and see if you get more replies.