Without beating through your working (which seems over-complex) we need to say that there is a solution to the Bessel equation such that C zero is non zero.
Once we have that solution written as a polynomial each coefficient in the polynomial must be separately zero.
i.e. if the solution is ax + bx^2 + cx^3 ... then a=0, b = 0, c=0 etc and each of a, b, c will be a polynomial in r.
So work out the bits (I write C rather than C sub zero to save having to latex this)
y=Cx^r, y'=Crx^(r-1) and y''=Cr(r-1)x^(r-2)
when these are substituted into Bessel's equn we get for terms involving C zero (again just using C). We are only interested in the C zero term
y(x) = Cx^r [ r (r-1) + r + (x^2 - m^2)] + other terms not involving C zero
Now multiplying this out (and rejecting the term in x^(r+2) cos that is not an x^r term) we get the coefficient of x^r is C[r^2 - r + r -m^2]
If C is non-zero then for the Solution to be valid the r term in the coefficient of x^r must be zero.
i.e. [r^2 - r + r -m^2] must be zero which gives us r^2=m^2
Hope that's clear