Generally, "cut in half" would be my first choice, but other situations may use "into halves. The only one that I can't imagine using is "cut into half.".
Cut the squash in half, scoop out the pulp, and place the two havles face up in a baking pan.
I cut the sandwhich into halves, taking one for myself and giving the other to Jamie.
If the description is just to show that it is cut into two pieces, I'd use "cut it in half."
If something will be done with the two halves afterwards - if they will be eaten, or taken away, or divided up somehow, then "cut into halves" sounds good.
There's not a huge difference. Again, don't use "cut into half" but the others are okay.