Your new drive will still be 60GB. If I remember correctly, the 10/20GB drive was due to a manufacturing issue--Toshiba didn't have 10GB drives for the F10s, but did have 20GB drives. So they fiddled to make only 10GB visible and shipped them out like that.
I would assume it's the hard drive, not the player, so a new SSD will be completely usable.
Be careful while swapping drives in the Gigabeat--the OF looks for certain files on the HDD, so follow the directions carefully. If you don't have an external HDD enclosure for your new drive, I'd recommend picking one up. They are pretty inexpensive, and very useful. Besides, when you're done, you'll have a bonus 10GB portable HDD.