Can anyone explain or push me in the right direction on how to show/answer this?
The first obvious thing to ponder is "what does it mean that a 'group is cyclic'? What properties does a cyclic group have, that noncyclic groups do not?"
(Broader hint: a cyclic group has a generator; noncyclic (finite) groups do not)
I understand that points of order two (from the hint) follow the formula:
2P = 0 but I'm not sure how this relates.
It relates quite a bit, actually. Can a cyclic group have two different elements of order 2?
Update: since Dexter has solved the problem, I'll lay out the proof I was originally trying to suggest.
One fundamental property of all finite groups is that, if $n$ is the number of group elements, we have $xP = (x \bmod n)P$ (for any integer $x$ and any element $P$).
In addition, a finite group is cyclic if it has a generator, that is, an element $G$ for which all elements can be expressed as $xG$ (for some $x$). In addition, for any $P$, we always have $xG = P$ for some $0 \le x < n$. In particular, $xG \ne 0$ for any $0 < x < n$.
Now, for the group in question, we have two distinct elements $P_1$ and $P_2$, both of order two (actually, we have three, we only need two). That is, $P_1 \ne P_2 \ne 0$, and $2P_1 = 2P_2 = 0$.
Now, suppose the group is cyclic; if so, we have a generator $G$. We know $P_1$ can be expressed as $P_1 = x_1G$ for some $0 \le x_1 < n$. We also have $2P_1 = 0$, that is, $2 (x_1 G) = (2 x_1)G = 0$. We know that $(2x_1)G = (2x_1 \bmod n)G$, and if that is the neutral element, we must have $2x_1 \bmod n = 0$. The only value $0 < x_1 < n$ that can satisfy that is $x_1 = n/2$ (which implies that $n$ must be even).
However, by the same reasoning, if we have $P_2 = x_2G$, we have $x_2 = n/2$, but that would mean that $P_1 = (n/2)G = P_2$, and so the two points of order two were the same, but we know they aren't.
Hence, there cannot be a generator, and hence the group cannot be cyclic.
This proof may sound a bit lengthy, however, that's mostly because I spelled out the logic quite explicitly, listing the various assumed properties of groups. Once you get more acquainted with group theory, this logic would be more natural.