I would remove one of the nails and verify that they are annular ring nails. If not, the problem will keep appearing. Regular joint compound will work just fine. If you have a problem with rust stains from the nail heads, then a little KILZ primer will rectify that. GOOD LUCK!!!

Thanks, but I think that you misinterpreted my question. The nail pops are just showing as bumps in the drywall. Do I remoe these popped screws or nails and put new screws in just above/below where the popped nail was removedd?

My apologies! They are showing as bumps for a number of reasons (you must determine the correct reason since I am not there). Either the nails were not set properly when the drywall was installed....you can hammer the nails back in if the backing behind the nails is not broken.
The nails are not annular ring....The nails must be removed and annular ring drywall nails or screws in a new location should be installed.
The wall studs are moving (yellow pine is a good example).... so the nails will have to be removed and new ones installed.
If you have any doubts then yes it is better to remove the nails or simply pound them back into the stud as far as they will go (a round punch the size of the nail head will cause less damage to the surrounding drywall thus less patching) and install new ones as you have described.
I hope that I have done a better job of answering your question this time. GOOD LUCK!

In my experience I have found that it is best to drive coarse thread drywall screws into the studs close to the nail pop and either drive the nail back in or remove it.