would i have to solder the switches back in?
I'll guess that IF (*) you're able to get spare parts, that will inclute both the little wire and the little pcb that the button itself are mounted onto so you won't have to solder. Again, the problem will be to actually get that spare parts and how much it will cost.
Alternatively, the button itsellf may or may not be possible to switch but it requires soldering. The question here is whether the button is spechial made for that particular mouse model/series or if it is a standard button. It is important question because the possibillity you'll have to actually be able to buy another one.
(*) Thing is - you may not be able to buy spare parts from the manufactor unless you're...