Let me answer the easy part now and then come back later for the (slightly) more delicate singular case: If $(M,J)$ is a real-analytic almost-complex manifold and $X,Y\subset M$ are almost-complex submanifolds of $M$ whose intersection $X\cap Y$ is a submanifold, then $X\cap Y$ is an almost-complex manifold.
To see this, recall the following easy consequence of elliptic theory: Suppose that $(M,J)$ is a real-analytic almost-complex manifold and that $u:\mathbb{R}\to M$ is a real-analytic curve in $M$. Then there exists an open neighborhood $W\subset\mathbb{C}$ of $\mathbb{R}$ in $\mathbb{C}$ and a map $h:W\to M$ that is pseudoholomorphic (i.e., $h'(iv) = Jh'(v)$ for all tangent vectors $v\in TW$) such that $h(x) = u(x)$ for all $x\in\mathbb{R}$. Moreover, $h$ is locally unique in the sense that any two such extensions $h_i:W_i\to M$ agree on some open neighborhood of $\mathbb{R}\subset W_1\cap W_2$.
This immediately implies the claim: Since $X$ and $Y$ are almost-complex manifolds and $(M,J)$ is real-analytic, they are real-analytic submanifolds. If $X\cap Y$ is a submanifold (necessarily real-analytic) and $v$ is a tangent vector to $X\cap Y$, then there will be a real-analytic $u:\mathbb{R}\to X\cap Y$ such that $u'(0) = v$. Let $h:W\to M$ be a pseudo-holomorphic extension as above. Since $u:\mathbb{R}\to X$ and $u:\mathbb{R}\to Y$ are real-analytic, and $J$ restricts to each to be an almost-complex structure, it follows by the local uniqueness that, by shrinking $W$ if necessary, it can be assumed that $h(W)\subset X\cap Y$. But now $h'(0)(\partial_x) = u'(0) = v$, so $h'(0)(\partial_y) = h'(0)(i\partial_x) = Jv$. Thus, $Jv$ is also tangent to $X\cap Y$. Thus, $J\bigl(T(X\cap Y)\bigr) = T(X\cap Y)$, so $X\cap Y$ is almost-complex.
Added remark about the singular case: I still don't have time to write out a careful argument for this (and my real-analytic singularity theory is a little rusty), but I thought I'd go ahead and put in the sketch of the argument for what happens when $X\cap Y$ is not assumed to be a submanifold (but I am still assuming that $(M,J)$ is real-analytic). The point is that $X\cap Y$ is still a real-analytic variety and, as such, has a Whitney stratification into a disjoint union of real analytic submanifolds (not necessarily closed).
Recall that the analytic tangent variety $\mathsf{A\!T}(X\cap Y)\subset TM$ is defined to be the closure of the set of vectors $v\in TM$ that are the velocities of real-analytic curves $u:\mathbb{R}\to M$ whose images lie in $X\cap Y$. The argument given above in the case that $X\cap Y$ is a submanifold clearly extends to the more general case, showing that $\mathsf{A\!T}(X\cap Y)$ is invariant under $J$. This fact, coupled with the properties of the Whitney stratification (i.e., that the WS satisfies Whitney's Conditions A and B), should show that each of the Whitney strata of $X\cap Y$ is an almost-complex submanifold of $M$, thus implying that $X\cap Y$ is a disjoint union of almost-complex submanifolds of $M$. I'm pretty sure that this is will go through, but it does appear that the details are somewhat tedious.
