Most times, squeaking is the belt itself slipping. If the outside (non-ribbed) part of the belt is slick and shiny, or has very shiny spots, it has slipped and needs to be replaced.
If the belt is fine, sometimes the cause can be the pulleys themselves. The pulley squeaking is usually the idler or tensioner pulley, and you can fix it by either replacing just the bearing inside the pulley (remove the belt, then unbolt the pulley, and then you can see the number on the bearing dust cover to buy a new one), or just buying a new pulley with the bearing already pressed in.
Sometimes squeaking can be a water pump going too, so be sure it's not that.
It's perfectly normal for the wheels to move a bit when going over bumps, especially large bumps or when going off road. When going over bumpy terrain, holding the steering wheel on the outer edge is advised in case the wheel turns quickly, so it doesn't catch your hand or arm and move it farther than you're prepared for.
Changing the fluid in the power steering with new Type F ATF and being sure the level is correct might help if there are issues or it feels like the steering isn't as controlled as you might like.