In a tweet from Jeremy Crawford, he says:

When a spell's description uses "enter" in relation to an AoE, the entering has to be voluntary only if the text says so.

By itself, it seems to imply that moving the Moonbeam counts as "entering".

But, in a Sage Advice article, he clarifies this:

Our design intent for such spells is this: a creature enters the area of effect when the creature passes into it. Creating the area of effect on the creature or moving it onto the creature doesn't count. If the creature is still in the area at the start of its turn, it is subjected to the area’s effect.

Entering such an area of effect needn't be voluntary, unless a spell says otherwise. You can, therefore, hurl a creature into the area with a spell like thunderwave. [...] Keep in mind, however, that a creature is subjected to such an area of effect only the first time it enters the area on a turn. You can't move a creature in and out of it to damage it over and over again on the same turn.

In summary, a spell like moonbeam affects a creature when the creature passes into the spell’s area of effect and when the creature starts its turn there. You’re essentially creating a hazard on the battlefield.

Under the default rules there are no squares and the caster is free to pick any point in 3D space; the 5-foot radius then extends from that point and if any creature is touched then it is affected.

I assume you are using the Playing on a Grid variant on page 192 of the Player's Handbook. In which case your caster still decides which point in 3d space to target and your DM decides which squares are affected.

Alternatively, you can use the variant to the variant rule on p. 251 of the Dungeon Master's Guide which says:

Choose an intersection of squares or hexes as the point of origin of an area of effect, then follow its rules as normal. If an area of effect is circular and covers at least half a square, it affects that square.

So your Moonbeam affects the 4 squares that touch the intersection. A 15 foot radius fireball affects 32 squares (a 6x6 area with the corners missing).