Based upon everything I have ever heard about the Fender Squire, you would be better off upgrading to another guitar than investing money into improving the sound of the Fender Squire. The Fender Squires that are manufactured today in Korea are quite a bit lower in quality that the Fender Squires which were originally manufactured in Japan. I know people who have bought Squires and really had trouble keeping them in tune. Plus the current Squires aren't made of a solid piece of wood (I believe they currently are made of a laminated form of particle board). Plunking down a lot of money to upgrade, the electronics, and hardware is only going to give you marginal improvement in the overall sound of the guitar.

However, you haven't been playing for all that long. If the Squire stays in tune and doesn't feed back excessively, you should think about taking lessons before upgrading to another guitar. When you get a little more playing experience, you will have a better idea what you want in terms of sound, guitar set up, etc... Plus, your lack of good sound could just be a symptom that you have hit a plateau with self teaching. Taking lessons and learning something new, could make your current set up sound better to you.