If you force creature into the Snare spell square does it activate?

Question

Snare is a spell that activates when

The trap triggers when a Small creature or larger moves into the area protected by the spell.

Would forced movement via the Thornwhip cantrip snare the foe?

Answer 1

Compared to similar rulings, yes

The Grasping Arrow from the Arcane Archer says:

The creature hit by the arrow [...] takes 2d6 slashing damage the first time on each turn it moves 1 foot or more without teleporting.

For that J. Crawford ruled, that forced movement counts as a move. Booming Blade on the other hand says the following:

If the target willingly moves 5 feet or more [...]

Unwilling moves are here specifically excluded. Consider that on page 7 of the PHB it says:

Specific Beats General

Which means the following: "Move" means every possible way of movement, willing or unwilling, by foot, air or water, except if it says something different.

Answer 2

Yes.

There are a number of spells in Fifth Edition which produce effects when a creature transits their areas of effect and this is one of them. The wording for each spell can vary a little, indicating whether or not the movement needs to be willing or not.

For example:

Prismatic Wall:

When a creature attempts to reach into or pass through the wall [...]

Wall of Fire:

A creature takes the same damage when it enters the wall for the first time on a turn or ends its turn there.

Snare says:

The trap triggers when a Small creature or larger moves into the area protected by the spell.

The Sage Advice Compendium includes a discussion on this topic in regard to a different AoE spell (Moonbeam):

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.

It is worth pointing out that they're talking about spells that have wording similar to Moonbeam which trigger when a creature "enters the area of effect". They don't include it on the list, but this is also how Snare is worded.

Sage advice summarizes it succinctly: You’re essentially creating a hazard on the battlefield. Snare is a magical trap, much like any trap that is built into the design of the adventuring environment.

If I push a monster onto a mundane pressure plate trap in a dungeon, I expect it to go off without consideration for whether or not the monster wanted to end up on that pressure plate.

To that end, Snare, once cast, becomes a trap like any other and any appropriately sized creature that ends up in the Snare's space, friend or foe, willing or unwilling, would trigger the trap.