IMO: B

Good problem.

Statement(1):
(x - y)(x + y) > 1
Since the product of two quantities is positive, either both of them are positive
for e.g.
x - y = 2 and x + y = 4
OR
both of them are negative
for e.g.
x - y = -2 and x + y = -4
INSUFFICIENT

Statement(2):
x/y + 1 > 0
x/y > -1
For x/y > 0, x and y are both positive (x + y > 0) or both are negative (x + y < 0)
We don't need the test the x/y < 0 case because this statement is already
INSUFFICIENT

Let's combine.
For x/y + 1 > 0 in Statement (2),
for both x > 0 and y > 0, as well as x < 0 and y < 0, there are values (when |x| > |y|) for which Statement (1) holds true.
So, even after combining x + y > 0 or x + y < 0.

[spoiler](E)[/spoiler] is the answer.

Thanks Mitch, realised my mistake after seeing your response.

karthikpandian19 wrote:Is x + y > 0 ?
(I) x² - y² > 1
(II) x/y + 1 > 0

If x=3 and y=1, both statements are satisfied.
In this case, x+y>0.

If x=-3 and y=-1, both statements are satisfied.
In this case, x+y<0.

Thus, the two statements combined are INSUFFICIENT.

The correct answer is E.