My sincerest apologies, I forgot to add the 90 degree marks
Without them, this question is CRAZY hard.
I need to pay more attention.

Answer: A

That's a good question, MSD.
We don't actually need to find the measures of BF and BE etc. We need only show that we could (i.e., we have sufficient data to do so) find the measures.

First, we'll show that ABCD is a square [without using (1)]

Since EFGH is a square, we'll let each side of the square be of length x
You have already deduced that all of the triangles are 30-60-90 triangles, and now we know that they all have a hypotenuse of lenght x.

This means that each of the 30-60-90 triangles has lengths x/2, sqrt(3)x/2, and x

This will show that ABCD has sides of equal length so it is a square.
Now, what is the length of each side of square ABCD?
It's x/2 + sqrt(3)x/2
But we already know the length of each side of ABCD. Since the area is 16, we know that each side is length 4.

This gives us the equation x/2 + sqrt(3)x/2 = 4

This is an equation (a linear equation) with one unknown, which we could solve if we were so inclined. Solving for x would give us the dimensions of the inner square (EFGH) so we could determine its area.

So, (1) is enough info