The error was in my transcription. That sign error was irrelevant because, no matter what I typed, I knew that the squares cancelled out.

Your trouble following the next step was due to another typo, 4 instead of 2. The idea goes like this:

Eqn 2 - Eqn 1:

  (x2 + y2 - 2y + 1)
    - (x2 - 2x + 1 + y2)
      = m(z2 - 2zh2 + h22)
        - m(z2 - 2zh1 + h12)

  -2y + 2x
    = m(- 2zh2 + h22 + 2zh1 - h12)

Now divide both sides by m. Do the same for the other equation.