Fender MIM Jazz, adding the Chrome bridge cover/Pickup cover?
 

Do any additional holes need to be drilled for either of these?

Generally yes, but some people use double sided tape instead, so they don't have to drill holes. I prefer using the screws, it looks better IMO.

Better to use screws. Make absolutely sure you have it in the right position before you drill holes to mount. If you mount with double sided tape, you will not have access to adjust string saddle height.