No, only the Squier (and other MIDI-guitars) will work.
From Game Informer, Issue 209 (pg 32):
"Due to technology in the neck the Squier does not need to be re-tuned to alternate tunings," Baker assures us. "No capos are needed to play the game." ... Baker says you don't need to be in tune to play the game.
...
"The Rock Band 3 Squier Stratocaster uses technology built into the neck and fingerboard of the guitar to track finger positions in real time," explains Baker. As a player holds down a particular fret and string, the onscreen display will show that note and string, giving players feedback on correct finger placement even before the note is picked.
What all this means is the game tracks where your fingers are and when you pluck, not the sound the guitar makes.
It's theoretically possible that an accessory could be made to play the game using only the sounds made by a guitar, but doing this is notoriously difficult (and impossible in truly real-time - see this page and @Jeffrey's comment below), and would likely be very inaccurate and frustrating.
I've looked all over for the converter mentioned by @jblaske in the comments above, but can't find anything. He is probably thinking of the Rock Band 3 MIDI Pro Adapter, used to play Rock Band 3 with midi-keyboards and midi-guitars (not the same as electric guitars!).
Best Answer
Yes, you can play both standard mode and the bass parts with the Mustang.
Frets 1-5 are used to simulate the Green->Orange buttons that you may be used to on the standard RB Strat, though they're wider. You can also use frets 13-17, which are narrower and liken a bit more to the solo buttons from the RB Strat, and they function in the same manner in which you don't have to strum for the solo/Big Rock Endings parts. You can press any or all buttons on the appropriate fret, it doesn't matter which 'string' the buttons are on, and similarly you can strum any string whether you're fretting it or not.
For the bass parts in Pro Mode, you can simply ignore the 2 strings you don't use (B and high E).