Does the Rock Band 3 Squier lock you into playing notes/chords a certain way for the game?

Yes, you have to play it the way the game tells you to. But I think you may misunderstand how the charts in the game are shown. It doesn't tell you "play an E", it says "play the 5th fret of the A string". Trying to convert what you're seeing to a completely different method of playing the same note, while it's scrolling towards you, would be extremely difficult.

I do remember reading that they put a great deal of effort into making sure that the charts in the game were the way the song was actually played in terms of location though, comparing with live videos of the actual bands playing the song and such. So if that's true you may be making it harder on yourself by trying to use alternate methods.