You could use the tangent of the angle of the dir vector. Depending on the direction it is pointing, you use a different formula based on the angle of the diagonal of the AABB. I've came up with this:

[source lang="cpp"]
// WARNING: Untested code
Point Intersect(Point min, Point max, float alpha) // alpha is the dir vector angle
{
float beta = atan2(max.y - min.y, max.x - min.x); // AABB diagonal angle
float hw = (max.x - min.x)*0.5; // AABB width/2
float hh = (max.y - min.y)*0.5; // AABB height/2
float tanAlpha = tan(alpha);
if (fabsf(alpha) < beta) // right section
return Point(max.x, min.y + hh + tanAlpha*hw);
else if (fabsf(PI - alpha) < beta) // left section
return Point(min.x, min.y + hh - tanAlpha*hw);
else if (fabsf(PI*0.5 - alpha) < PI*0.5 - beta) // top section
return Point(min.x + hw + 1.f/tanAlpha*hh, max.y);
else if (fabsf(1.5f*PI - alpha) < PI*0.5 - beta) // bottom section
return Point(min.x + hw - 1.f/tanAlpha*hh, min.y);
else // should not happen o_ O
return WTF;
}
[/source]