The guage and quality of the wire makes absolutely ZERO difference in the tone of the guitar. If your guitar hums or picks up interference (and it's NOT from the single coils), then switching to shielded wire, tidying up the mess of wire, or shielding the whole ******* cavity (ala the "quieting the beast" article) would improve it. Probabally a moot point if you have non-hum-cancelling (i mean "classic") signel coils. The quality of the pots/switches/jacks won't effect the tone (but the VALUES will!), but should be quality parts anyways. The cheap Squiere pots are not durable and will get scratchy pretty quickly. Quality pots will last for a long time before they start to get scratchy, are harder to break, and generally have a very good feel when you turn them. The squire uses a PCB switch. I heard that these wear out quickly and that the actual wafer switch, like what is used in a real fender, is supposed to last for a decent amount of time. A good quality input jack (you are looking for solidly constructed, not necessarily 24K gold plated) should last longer before it breaks. They also typically hold the cable better and you won't have to jiggle it (like my Friend's squier P bass....) to get it to work after a few months. Again, the cheap parts will not negatively effect the sound, just the reliability,"smoothness" of control, and pops/buzzes/shorts.
The capacitor off of the tone controls is somewhat important. THis capacitor shou be a film type and not a chepa ceramic. Remember that the glossy green (sometimes red) squarish capacitor that looks like a Chicklet is the good kind, and the flat round orangy-yellow one that looks like a smooshed booger is the bad one. The rating in uF of this cap is critical to the sound, and I highly recommend that you get different values and play around with them. Voltage rating and brand here is unimportant.
ANd the most important electrical part, much much more important than anythign I've discussed so far, would be pickups and the amplifier. what's in your fingers trumps everything else, however!