Hell, even a standard Fender Strat is an option, they're only like $350US. And I don't care how much people knock the Mexican-made Strat, it can actually be a quality guitar if you know what you're doing with it. Just add 2 more springs to the 1-way locking tremolo (she only comes with three, which is a goddamn travesty), and you have absolute stability. Of course, the whammy bar will wear out faster if you do, and since the tremolo isn't a Floyd Rose double-locking unit, you'll have tuning problems if you do a Hendrix-style divebombing run.
Another option is the Arbor AP230. It's basically a poor man's ($350US) Les Paul, but it sounds just as good and is in fact easier to play (the body isn't as thick, so you get a slightly brighter tone and you can actually reach all the frets with relative ease). It comes in white w/gold hardware, black w/gold hardware, or cherry sunburst w/nickel hardware.
Unfortunately, bucky_2300 is very much correct in saying that a true upgrade is going to set you back a pretty hefty sum. But I'll be damned if it isn't worth every penny. And since you started on a Squier, you will most definitely notice all sorts of wonderful things.
Edit: And oh my god when you get your new guitar, if you don't know how to intonate your bridge, TAKE IT TO A PROFESSIONAL. If you must, practice on your older guitar, and ask questions before you do anything at all. But for the love of god don't make the mistake so many people make by complaining that the guitar wasn't worth the money they paid for it. It really was, they just didn't know how to intonate their bridge, and ignorance most definitely is not bliss when you're putting over a grand on the line.