Toward the end of "Characteristic Classes and Homogeneous Spaces, III," Borel and Hirzebruch prove that given a compact Lie group $G$ and toral subgroup $T$ (no restriction on rank), one has $w(G/T) = 1$ and $p(G/T) = 1$: the Stiefel–Whitney and Pontrjagin classes are trivial. In fact they show the tangent bundle of $G/T$ is stably trivial.
They say a similar statement is valid for Chern classes in the almost complex case. I guess that means that whenever $G/T$ is almost complex, $c(G/T) = 1$. Is that right? I suppose it's clear that if the underlying real bundle of a complex vector bundle is stably trivial, then the complex vector bundle is also stably trivial; just add a higher (even) number of trivial real line bundles.
I don't know anything about almost-complexity. I do know that $U(n)/U(1)^n$ is a complete flag manifold $Fl(\mathbb{C}^n)$, and from induction on fiberings $Fl(\mathbb{C}^{n-1}) \to Fl(\mathbb{C}^n) \to \mathbb{C}P^n$, that its Euler characteristic is $n!$, meaning its top Chern class is nonzero if it is defined. So, working backwards, I guess $Fl(\mathbb{C}^n)$ isn't almost complex, though I would have thought it should be. I don't know what happens to the other Chern classes even in this case.
I do know that if $T$ is not a maximal torus, then the Euler characteristic of $G/T$ is zero. But in this case, too, I don't know anything about Chern classes. Or maybe there's nothing to know. Suppose $G$ is a compact Lie group and $T$ is a nonmaximal torus. Can $G/T$ admit an almost complex structure?
They aren't actually referring to an almost complex structure, but to a real-stable version of it.
On the top of pg. 494: A bundle $\xi$ is called complex if $\xi \oplus 1^n$ (where $1^n$ is the $n$-dim trivial real bundle) has a complex structure for some $n$.
For example, if $\xi$ is the tangent bundle of $S^2$, then $\xi+1 = 1^3$ is trivial and so $\xi + 1^2 = 1^4$ has a (trivial) complex structure. It follows that $c(\xi) = 0$.
On the other hand, for the complex manifold $S^2$, $c(TS^2) \neq 0$.
(Note that $S^2 = U(2)/U(1)^2$, so this is directly relevant to your question.)
Finally, to answer your last question, if $G = SU(2) \times T^2$ and $U = S^1\times e\times e$, then $G/U = S^2 \times T^2$ so has a complex structure. I don't know of any less trivial examples, however.