Which is more electronegative: hydrogen or astatine?
An interesting way to determine which of $\ce{A}$ or $\ce{B}$ is more electronegative is to take their binary compound $\ce{A_xB_y}$ and observe its dissociation. Note that it can be $$\ce{A_xB_y -&gt; xA^y+ + yB^x-} \tag{1}\label{a}$$ or $$\ce{A_xB_y -&gt; yB^x+ + xA^y-} \tag{2}\label{b}$$

Reaction $\ref{a}$ indicates that $\ce{A}$ is less electronegative than $\ce{B}$, because in the dissociation of $\ce{A_xB_y}$, $\ce{A}$ carries the positive charge, while $\ce{B}$ carries the negative charge. Reaction $\ref{b}$ instead indicates that, for similar reasons, $\ce{A}$ is more electronegative than $\ce{B}$ 1



Going by the same logic, we may tend to observe a hydrohalo acid in this case, specifically, the hydroastatic acid/hydrogen astatide. Quoting from its Wikipedia page,


  This chemical compound can dissolve in water to form hydroastatic acid, which exhibits properties very similar to the other four binary acids, and is in fact the strongest among them. However, it is limited in use due to its ready decomposition into elemental hydrogen and astatine (my emphasis) Because the atoms have a nearly equal electronegativity, and as the $\ce{At+}$ ion has been observed, dissociation could easily result in the hydrogen carrying the negative charge. (my emphasis) Thus, a hydrogen astatide sample can undergo the following reaction:
  $$\ce{2HAt -&gt; H+ + At- + H- + At+ -&gt; H2 + At2}$$


which also explains the spontaneous decomposition into elemental hydrogen gas and the astatine precipitate. 

If you also have a look at their electronegativities, both have a Pauling electronegativity $2.2$, further complicating the comparison.

Therefore, based on these evidences, I can say, without the least hesitation, that theoretical comparison of the electronegativities of hydrogen and astatine is not possible.



[1]: a shrewd observer would notice that the apt way to write the second reaction's reactant would be $\ce{B_yA_x}$ instead, since $\ce{A}$ is more electronegative