lm(A~B2*C)
or
lm(A~I(B^2)*C)
give you the result of squaring column B and then doing the regression. Using
poly(B,2)
does something completely different - see ?poly.
Edit to add:
poly()
calculates orthogonal polynomials which are not the same as the standard polynomials derived from simply squaring, cubing etc. a number.