Sorry for OLD answer

First ask a simpler question: What is the maximum number of points of intersection for a line and a square? Because the square is convex, the answer is 2.

A Triangle consists of 3 line segments, each of which is a subset of a line. So each segment can clearly not intersect the square more than twice. This gives 6 as the upper bound, even if you extend each edge of the triangle into an infinite line.

Further note that if both endpoints of an edge are inside the square, then the whole edge is inside the square and doesn’t intersect it at all. And if one endpoint is inside and one is outside, then there can only be one point of intersection.

Therefore, you can only get 6 points of intersection if all 3 corners of the triangle are outside the square. If one is inside, the max is 4; if two are inside, the max is 2; and if 3 are inside, the max is 0.

Sorry for OLD answer

First ask a simpler question: What is the maximum number of points of intersection for a line and a square? Because the square is convex, the answer is 2.

A Triangle consists of 3 line segments, each of which is a subset of a line. So each segment can clearly not intersect the square more than twice. This gives 6 as the upper bound, even if you extend each edge of the triangle into an infinite line.

Further note that if both endpoints of an edge are inside the square, then the whole edge is inside the square and doesn’t intersect it at all. And if one endpoint is inside and one is outside, then there can only be one point of intersection.

Therefore, you can only get 6 points of intersection if all 3 corners of the triangle are outside the square. If one is inside, the max is 4; if two are inside, the max is 2; and if 3 are inside, the max is 0.

We might start by examining the number of ways that ONE SIDE of a triangle can intersect a square. 
In other words, in how many ways can a LINE intersect a square?
After a bit of mental imagery, we might conclude that a SINGLE LINE can intersect a square in at MOST 2 ways

A triangle is composed of THREE LINE SEGMENTS.
If each SINGLE LINE can intersect a square in at MOST 2 ways, then the 3-sided triangle can intersect a square in AT MOST 6 ways (with 2 intersections per line) 
So, the correct answer must be 6 or less

At that point, if we're able to sketch a scenario in which there are 6 intersections, we can be certain that this is, indeed, the GREATEST number of intersections.