A few things:
1) The traditional back squat is NOT a quad dominant excersize. It uses your entire body and if you can't get weight up it's most likely core weakness, not leg weakness.
2) Check your form. You're probably holding the bar incorrectly if your traps/shoulders are hurting. One mistake I see EVERYONE making at the squat rack is they grip the bar with their thumb under it and hold it wide. Powerlifters can get away with this because they have monster traps, but normal people can't. You want the bar to be resting on muscle and muscle only. To do this you have to "pinch" your shoulder blades together. This creates a cradle for the bar. For this to happen you need to take as narrow a grip as your flexibility allows, and put your thumb OVER the bar. Try this with just the bar first.
Your hands do NOT hold the bar like people think. They push it forward while your tensed traps push it back. This creates an equilibrium and holds the bar in place. Your hands actually have minimal involvement. I've seen videos of a guy squatting 315 for reps without even holding the bar - it was resting entirely on his traps. I don't recommend anyone doing this ever as this was done by a professional, but it goes to show where the weight should be distributed. I'm up to 270 for reps now and I've never had to use a pad and never had trap or shoulder pain.
If you don't get my explanation I suggest you pick up a copy of Starting Strength by Mark Rippetoe from
http://www.startingstrength.com or Amazon. Then again I think anyone serious about lifting should get it, but that's just me.