When a spell played with flow is countered, does it get banished instead of going to the trash?

Yes — a spell played for its Flow cost gets banished when it's countered, not trashed.

Flow's "then banish it" is a delayed replacement effect: if the spell would leave the chain after becoming a finalized chain item and the departure wasn't instructed by its own execution, it's banished instead (Rule 829.1.b.1). Being countered is exactly that kind of departure, so the trash destination is replaced by banishment — this is an explicit exception to the normal countered-spell rule (Rule 425.1.a.1). The same holds if the counter effect would otherwise return it to hand, e.g. Abandon (FAQ #11738).

Note the distinction: this only applies to the Flow play. A copy of that same spell played from hand for its normal cost and countered just goes to the trash.