I think it's technically possible to put eight of the cogs from a 9 speed cassette onto a seven speed freehub to make it an eight speed, but otherwise a seven speed freehub won't accept a normal 8+ speed cassette. And yes, in either case you would need a different shifter.

Generally speaking, you'd need a new rear wheel with an 8/9 speed hub, a new cassette, and the appropriate new shifter.