Must a target's shields be down in order for drones to do repairs?

Shields don't stop physical collisions with stuff, like ramming into stuff. I believe the lore of the shields is they just reduce the velocity of objects.

It's like hitting something into water, all of a sudden it slows down.

Yeah, so that's why energy weapons sap shields: they have no mass. So they don't go through the shield because they've got no mass to slow down. It just all gets absorbed in, which in turn draws down the shields, whereas ballistics go through shields because they've got the mass to carry on, and they go much faster. So a drone going slowly through it should be fine.

So I guess a shield wouldn't stop an EVA?

No, you can EVA through those.