Whenever you square both sides of an equation, you might be introducing false solutions, where the two sides of the equal have opposite signs. That's why at the end of this type of manipulation you need to verify that the values you found are actually solutions.
Squaring both sides of an equation is valid; squaring the terms of both sides is not, however.
But that's not what he did... He correctly deduced `1 = 49 * x' from `-1 = 7 * sqrt(x)'.
No, that's not a correct deduction. That's squaring each term, not the squaring entire side. When you square each term like that, you are assuming that ?(zw) = ?z?w, which is not generally true.
He said he squared each side, but then squared each term, not each side. That's the issue he had.