Why the Bears would take a kicker or punter: Reasons the Bears would take a kicker or punter in this year's draft would probably start with some illegal parties, drunken bets, or just sheer Eye-Dee-Ten-Tee errors on the part of Phil Emery. Seriously, I don't see a reason for the Bears to do so in this draft, with Robbie Gould still being really good and Adam Podlesh still being both well paid and pretty good. Maybe if they're concerned with Gould's leg at all after finishing last season on injured reserve or concerned with Podlesh's cap hit, but they don't seem like issues for right now.
As you'd expect, the kicker/punter discussion usually centers on whether the player's draftable at all, much less which round the player's expected to go in. Florida's Caleb Sturgis is almost a certainty to go in the mid-to-late rounds, but after that, not a whole lot else in this draft, with maybe Dustin Hopkins of Florida State and Quinn Sharp getting looks - the former having a really strong leg with some accuracy (might be more a kickoff specialist) and the latter being able to both kick and punt.
The rest of the class could be limited to undrafted tryouts, with Matt Weller having to show some accuracy to make a place and Brett Maher having limited experience.