What BB to get?

Just as an extra i use a Hope stainless bottom bracket on my mountain bike and its been excellent.Shimanos are cheaper but if you do wear out the bearings in HT2 bottom brackets you can replace the bearings without having to buy complete new units.RaceFace bearings used to be made of putty and lasted a couple of months.

There has to be threads in the bottom bracket for a square taper or octalink shaft.
Usually Shimano chainsets come with the bottom bracket cups if you're buying hollowtech 2.Not many chainsets on the market now that are square taper,which i think is a shame because i find they last a lot longer than the outboard bearings.
You'll need the splined tool to remove the old square taper bottom bracket and as dave said drive side clockwise to remove if English.You'll also need a screw in puller to draw off the crank arms first.
The Hollowtech 2 is dead easy to fit but another tool is needed to finally tighten in the bearing cups.
If you don't fancy buying the tools its an easy and usually quick job at a bike shop as long as the old bottom bracket isn't seized in.

It seems to me the original poster doesn't have threads for BB cups on his bike, so that being the case, what kind of BB will they need?

My via Nirone is English thread.

Yes you'll need a Hollowtech II compatible bottom bracket

If your drive-side thread on the bottom bracket is reversed (ie clockwise to remove) then it's english, if they're standard threads on both sides it's italian. i don't know which your frame will be i'm afraid.