What's going on here, I think, is you're confused about what you're being asked to prove.
The statement is "every odd integer is the difference of two squares", or, more precisely,
"for all odd integers $n$, there exist $a$ and $b$ such that $a^2 - b^2 = n$". Think for a bit about what the difference between "for all..." and "there exists..." is, and maybe you'll realise what's going on.
You have to come up with a proof that works for every odd integer, so you start with an arbitrary choice $n = 2k + 1$. But then you seem to go on to pick an arbitrary difference of two squares, $a^2 - b^2$. But that's not necessary – we're not trying to prove anything about every difference of two squares, just one, so you only need to show that there is a difference of two squares that satisfies the condition, and in particular you choose it, it isn't arbitrary.
Think of the statement as "if you give me an odd integer $n$, I can give you $a$ and $b$ such that $a^2 - b^2 = n$". In particular, you can give me whatever odd integer you like, but then I get to choose what $a$ and $b$ are. In particular, I can choose $a$ to be $b + 1$ if I like.
From one of your comments:
We have to show that $a^2 - b^2$ is an odd number
But in general that's just not true. For example, $4^2 - 2^2 = 12$. So if you've found yourself trying to prove that statement, you've clearly gone wrong somewhere.