Sum of Squares of Alternating Coefficients

If $c_0,c_1,c_2,\dots ,c_n$ are the coefficients in the expansion of $(1+x{)}^n$ for a positive integer $n$, prove that $c_0^2-c_1^2+c_2^2-c_3^2+\cdots +(-1{)}^n{c}_n^2=0$, or $(-1{)}^{\frac{n}{2}}{c}_{\frac{n}{2}}$, according as $n$ is odd or even.

H. S. Hall & S. R. Knight, Higher Algebra (1891) · Ch. XIV. Binomial Theorem: Any Index, Examples XIV.c, No. 18 (part 3)