What day is Nintendo’s eShop updated on

The short answer is no, you can't transfer individual eShop games.

From Nintendo's 3DS General Information page:

Can the downloadable version of a Nintendo 3DS game be transferred to another 3DS console?

It is not possible to transfer a specific software title to another Nintendo 3DS console. However, it is possible to perform a System Transfer and transfer the entire content of your console to another. For more information please refer to the System Transfer section of this Website.

I had to dig a bit to find this information, it is under 'Nintendo eShop Information -> Downloadable Retail Software - Frequently Asked Questions'. Also keep in mind this tidbit regarding System Transfer, under 'Transferring Games':

How Many Times Can I Transfer Content from One System to Another?

Data can be transferred up to five times. A system cannot perform a system transfer multiple times within a set period.

Cons of Downloaded Games:

• Can't share with friends
• Difficult to move from system to system
• Leaves cartridge slot empty (no placeholder)

Cons of Physical Cartridges:

• You are unable to back-up data into an SD card
• Need to carry/swap cartridges
• No possibility of recovery*

^{* Never assume Nintendo will get your digitally downloaded games back either. That being said, anecdotal evidence:}
I recently purchased a 3DS and purchased and downloaded a game. A week in, my hinges began loosening, so rather than sending it in for repair, I wanted to replace it at the retailer. I called Nintendo support, who told me that if I made sure to register my current 3DS first, and write down my serial number, they would be able to initiate a system transfer remotely, after I had exchanged my system. They said this is a new option, and I think it seems to be a step in the right direction. I swapped my system, and called them back. Unfortunately, their phones have been too busy to even put me in a queue for the last several evenings, so I can't yet confirm that this will in fact work, and am now left with a 3DS and none of my downloaded games. For now, this is definitely another pro for cartridges.

Really, it comes down to security of a physical copy vs convenience of a digital one. Personally it's physical cartridges for me from here on out, until Nintendo moves to a real account-based system.