I'm doing this also. The notches/splines in the freewheel body have nothing to do with the attachment to the hub, just removal. All freewheels thread onto the rear hub, and are tightened by pedaling. Unless you have an old French or Italian bike, any freewheel will have the same threads as yours. It will probably take a LOT of effort. Be sure to install an axle nut or QR skewer to hold the removal tool in place as you break the old freewheel loose. Don't forget to loosen the nut/skewer as you unthread the freewheel. The new freewheel threads back on hand tight. It will tighten more in use.

I'm not familiar with the Nashbar unit. I just ordered a NOS Suntour Ultra 7 to replace my original 6-speed. Holding them up to each other, the 7-speed is 1-2mm wider than the 6-speed, but it looks like it will fit. My bike has 126mm rear dropouts, which are probably the minimum for this swap. Hope this helped.

Regards,
John.