Let $g(x) = x^2 - 11x + 30,$ and let $f(x)$ be a polynomial such that
\[g(f(x)) = x^4 - 14x^3 + 62x^2 - 91x + 42.\]Find the sum of all possible values of $f(10^{100}).$
Let $d$ be the degree of $f(x).$  Then the degree of $g(f(x))$ is $2d = 4,$ so $d = 2.$

Accordingly, let $f(x) = ax^2 + bx + c.$  Then
\begin{align*}
g(f(x)) &= g(ax^2 + bx + c) \\
&= (ax^2 + bx + c)^2 - 11(ax^2 + bx + c) + 30 \\
&= a^2 x^4 + 2abx^3 + (2ac + b^2 - 11a) x^2 + (2bc - 11b) x + c^2 - 11c + 30.
\end{align*}Comparing coefficients, we get
\begin{align*}
a^2 &= 1, \\
2ab &= -14, \\
2ac + b^2 - 11a &= 62, \\
2cb - 11b &= -91, \\
c^2 - 11c + 30 &= 42.
\end{align*}From $a^2 = -1,$ $a = 1$ or $a = -1.$

If $a = 1,$ then from the equation $2ab = -14,$ $b = -7.$  Then from the equation $2cb - 11b = -91,$ $c = 12.$  Note that $(a,b,c) = (1,-7,12)$ satisfies all the equations.

If $a = -1,$ then from the equation $2ab = -14,$ $b = 7.$  Then from the equation $2cb - 11b = -91,$ $c = -1.$  Note that $(a,b,c) = (-1,7,-1)$ satisfies all the equations.

Therefore, the possible polynomials $f(x)$ are $x^2 - 7x + 12$ and $-x^2 + 7x - 1.$  Since
\[x^2 - 7x + 12 + (-x^2 + 7x - 1) = 11\]for all $x,$ the sum of all possible values of $f(10^{100})$ is $\boxed{11}.$