Also, only 2 of the 4 balljoints actually bear the weight of the vehicle, not ll 4, and I have no idea what sway arms are involved when making a turn, sway bars keep the vehicle level and are not really needed.
If it were mine I would take it to a trusted mechanic and have him test the ball joints, some have built in wear indicators, but who knows, you are asking for help and don't have enough sense to give us all the facts that are needed to give you an honest opinion . . . GOOD LUCK!!!
I have run into this many times. The squeaking is from a lack of grease in any one position. It could be as easy as getting under the vehicle with a pair of pliers, gripping the ball joint housing at the bottom and twisting it back and forth from front to back and redistributing the grease inside the boot. If the boots are not torn and it has not been squeaking for an extended amount of time this might work.
Ball joints support the weight of the vehicle and allow the wheel to turn. If the rubber boot cracks and the grease leaks out they wear. It's a ball and socket arrangement, so both the ball and socket wear until the ball finally pops out and the wheel collapses. They have to be pretty worn to make noise.
So yes, they're a major safety item. The sway arms prevent the opposite wheel from lifting off the ground during turns.
Since you didn't tell us the year, make, model of this mystery vehicle, no one can tell you if the price is fair. But you should get a second opinion to see if you really need them and to see if the price is fair.
Ball joints have nothing to do with pulling that trailer. Ball joints don't fail unless they are REALLY worn out, plenty of cars on the road with worn out ball joints every day. Worry about that after you get there. Plenty of people are fooled by incorrect diagnoses of bad ball joints of course the mechanic won't tell you that.
Looking over the procedure, I can see that with labor costs it would run you close to that; assuming that the upper ones need to be replaced. Those are the ones that take some time as they are pressed into the upper A-arms. Either replace the entire A-arm, which is what I would do since they need to come out anyhow, or use a hydraulic press to change the ball joints.
1st get second estimate. If you need ball joints dont drive without fixing. Could be dangerous
Myself and a friend are moving, and i took my truck to get it checked to see if it could pull a uhaul trailer (not a large one, it's 4'x8'). The mechanic said the ball joints and sway arms need to be replaced because it squeaks when i turn, and it would cost $1,000+. Do i absolutely need this to go to Maine?