When you control a squig herders it says that the unit gains one power if you control one damaged unit. If this unit is either on quest or kingdom will the power acquired through the damaged unit allow you to gain more resources (kingdom) or perhaps gain an extra card (quest) per turn?

Sure. When power is counted, like during your kingdom and quest phases, it doesn't matter where it comes from, if it's power printed on the card, power gained from a constant effect like Squig Herder's, or from a temporary one like Warpstone Experiments.