My iPhone 4 (model AI332) won't turn on or charge My iPhone's back got smashed and, not liking the look of it, I took it off and used the phone without the back. 

Later, I dropped the phone again and the battery came out completely. Although the battery was a bit loose before I dropped it the second time, now it wasn't even in the phone.

I put the battery back in and because it would not turn on, I began charging it. The iPhone started booting up, but then it shut off and restarted. It did this several times before it finally brought up an image showing me to plug it into iTunes.

When I did connect it to iTunes, nothing happened; the iPhone did not light up at all as I plugged it into my laptop.
 I went to a phone repair store, and a gentleman told me it was the battery, but is there any way I can fix this myself and is it definitely the battery?
The real answer is you can:

a) pay someone to test a new battery in the device and tell you if it's just the battery
b) buy a new battery (or one that is so old and used, that it doesn't hold a long charge, but works well enough to restore the device when it's being powered from a computer).

There's no way the internet can do the diagnosis on one device with the details you have provided. We can explain how to test things, how things work but generally refrain from making guesses.