Not an answer, but this may help with asymptotics:

According to Maple the o.g.f. for $H^s_n$ is

$$ \sum_{j=1}^\infty j^{-s} (-1)^{j-1} \sum_{n=j}^\infty {n \choose j} x^n = {\frac {1}{-1+x}{\it polylog} \left( s,{\frac {x}{-1+x}} \right) }$$

In particular this should be analytic for $|x|<1$.

Not an answer, but this may help with asymptotics:

According to Maple the o.g.f. for $H^s_n$ is

$$ \sum_{j=1}^\infty j^{-s} (-1)^{j-1} \sum_{n=j}^\infty {n \choose j} x^n = {\frac {1}{-1+x}{\it polylog} \left( s,{\frac {x}{-1+x}} \right) }$$

In particular this should be analytic for $|x|<1$.

EDIT: For positive integer values of $s$, Dilcher's formula says $$ H_n^s = \sum_{1 \le i_1 \le i_2 \le \ldots \le i_s \le n} \dfrac{1}{i_1 i_2 \ldots i_s} $$

Not an answer, but this may help with asymptotics:

According to Maple the o.g.f. for $H^s_n$ is

$$ \sum_{j=1}^\infty j^{-s} (-1)^{j-1} \sum_{n=j}^\infty {n \choose j} x^n = {\frac {1}{-1+x}{\it polylog} \left( s,{\frac {x}{-1+x}} \right) }$$

Not an answer, but this may help with asymptotics:

According to Maple the o.g.f. for $H^s_n$ is

$$ \sum_{j=1}^\infty j^{-s} (-1)^{j-1} \sum_{n=j}^\infty {n \choose j} x^n = {\frac {1}{-1+x}{\it polylog} \left( s,{\frac {x}{-1+x}} \right) }$$

In particular this should be analytic for $|x|<1$.