If I play Rengar, Trophy Hunter as a reaction when my opponent moves a unit into a battlefield, do Assault or Shield abilities still trigger?

Assault and Shield are both passive abilities (not triggered abilities) — they don't "trigger" at all. They apply continuously while their conditions are met. But the timing matters here.

During the non-combat showdown (while the chain is resolving your Rengar play and before combat begins): Neither applies. Combat hasn't started yet. Assault requires being "an attacker during Combat" (Rule 807.1.d) and Shield requires being "a defender during Combat" (Rule 814.1.d). During the initial non-combat showdown, the Attacker/Defender designations exist (FAQ #10937), but no combat is ongoing, so the keyword conditions aren't satisfied.

Once combat officially begins (after the chain empties and both players pass focus): Both apply. The opponent's unit retains the Attacker designation (Rule 459.2.b) — if it has Assault, it gets +X Might. Rengar retains the Defender designation — if he had Shield (he doesn't, but the principle stands for any defending unit), he'd get +X Might.

In short: Assault and Shield are dormant during the non-combat showdown phase, but kick in as soon as combat starts and the designations carry over.