Given a nonseparable compact Hausdorff space $X$ must the stone-cech boundary of the naturals, $\beta\mathbb{N}\setminus\mathbb{N}$ embed into $X$?
I admittedly don't have too much intuition regarding this. I believe that if $X$ itself appears as a stone-cech boundary of a Tychonoff space then the answer is yes (though I haven't written it down). However the general question seems more difficult.
No, e.g. ordered compact non-separable Hausdorff spaces, like $\omega_1+1$, or the lexicographically ordered square cannot contain the Cech-Stone remainder of $\mathbb{N}$ (the latter is not hereditarily normal, the former are). I think the Alexandrov double of the $[0,1]$ or the circle is also an easy example (an embedding misses the isolated part, and must lie in the metric part, contradiction). There are probably more easy examples.