Hey, I have an old fender squier and was wondering. If I upgrade the pickups, bridge and neck, could the overall quality possibly be increased to that of a guitar in the 1000 dollar range ?
The quality of the instrument, no. But you can improve the hardware, electronics and pickups to much higher quality ones. You are stuck with the cheap wood of the guitar.
2002 PRS CE22
2013 G&L ASAT Deluxe
2009 Epiphone G-400 (SH-4)
Marshall JCM2000 DSL100
Krank 1980 Jr 20watt
Krank Rev 4x12 (eminence V12)
GFS Greenie/Digitech Bad Monkey
Morley Bad Horsie 2
MXR Smart Gate
Does that mean that the type of wood the body is made from has an effect on the sound and/or feel?
Yes. The type and the quality of the wood.
For the most part, by the time you're done buying all of that stuff (especially the neck!) you could have just gotten a better guitar. Of course, then you'll want to upgrade that eventually... but you'll have a better platform. Plus you could start with something good instead of throwing money at what might never be a great instrument.

Some of those old Squiers were rather good guitars - but a lot weren't. I wouldn't sink a bunch of money into a Squier unless you already are very sure that the body and general fit are very good.

Short version - upgrade good instruments to great ones. Don't upgrade bad ones into mediocre ones, or expect to upgrade from mediocre to great.