All of the solutions for $F_{n + 4m} - F_n$ are the ones listed, barring the cases $m=2$ and $m=12$ where I still have some kinks to work out.

Following Will Sawin's suggestion, we write $F_{n + m} - F_n$ as a reccurence, with some initial terms.
Taking $n=0$ tells us that $F_m$ is one term, and $F_{m+1} - 1$ is the next.
Extrapolating backwards, we have the general term $F_{m-i} + (-1)^i F_i$.
So we see that if $m=4k+2$, then the term $i = 2k+1$ gives us $0$, and the next term gives us $F_{2k} + F_{2k+2} = L_{2k+1}$, so we have a copy of the Fibonacci numbers, multiplied by $L_{2k+1}$.
That is, $F_{n + 4k+2} - F_n = L_{2k+1}F_{n + 2k+1}$.
If $m = 4k$, then the term $i=2k$ gives us $2F_{2k}$, and the next term gives us $F_{2k+1} - F_{2k-1}=F_{2k}$, so we have a copy of the Lucas numbers, multiplied by $F_{2k}$.
That is, $F_{n + 4k} - F_n = F_{2k}L_{n + 2k}$. This is useful, because there are certain primes which never divide Lucas numbers (see http://oeis.org/A053028), so if $F_{2k}$ contains one of these primes (exactly one is easiest, but any odd multiple will do), then the product cannot be a square.

Agol gave a link to a paper in the comments (http://www.fq.math.ca/Scanned/7-1/ferns.pdf) which also contains this result.

Let's find all squares where $m = 2^k$. The cases $k=0, 1$ were given by AH in the comments, and the case $k=2$ is also included there.
We need two facts about Lucas numbers, both of which are easy to show. First, $3 | L_m$ iff $m \equiv 2 \pmod{4}$ and second, $7 | L_m$ iff $m \equiv 4 \pmod{8}$.

Since $4 | 2^k$, we can write $F_{n + 2^k} - F_n = F_{2^{k-1}}L_{n + 2^{k-1}}$.
I have not resolved the case $k = 3$ (this gap has been fixed, see 2nd edit below), it is equivalent to finding Lucas numbers $x$ in the Diophantine equation $x^2 + 2 = 3y^2$. For now assume $k > 3$.
By the facts above, we know that $3$ and $7$ cannot both divide a Lucas number, so we want to show that each of them divide $F_{2^{k-1}}$, and that neither $3^2$ nor $7^2$ do so.
That both 3 and 7 divide follows from $F_8=21$ and $F_n | F_{2n}$.
That both 9 and 49 don't divide follows by induction: the base case is $F_8$; and we have $F_{2^{k+1}} = F_{2^k}L_{2^k}$, and neither 3 nor 7 can divide $L_{2^k}$ when $k>2$.
Then by induction the squares do not divide $F_{2^k}$ for $k \geq 3$, and neither $3$ nor $7$ divide $L_{2^k}$, since $2^k \equiv 0 \pmod{8}$.
Hence there are no squares when $m = 2^k$ and $k > 3$.

For the general case of $m=4k$, we write $F_{n + 4k} - F_n = F_{2k}L_{n + 2k} = F_{k}L_{k}L_{n + 2k}$ and invoke Carmichael's theorem: for each $n>3$, there is at least one prime $p | F_n$ which divides no previous Fibonacci number. Such a prime is called a primitive.
Further - this is not part of the theorem - $p^2 \nmid F_n$ (after trying to work out a proof of this, I went to the literature and found that this is a conjecture in P. Ribenboim "Square classes of Fibonacci and Lucas numbers" Port. Math 46 (1989), 159-175. I'm not sure if it's been proven or falsified since then). Taking $n$ to be odd, we exploit the fact that $p$ does not divide any Lucas number, since its Fibonacci entry point is odd (see C. Ballot and M. Elia, "Rank and period of primes in the Fibonacci sequence; a trichotomy," Fib. Quart., 45 (No. 1, 2007), 56-63).
If the odd part of $k$ is greater than $3$, we are done, since we can continue splitting $F_{k} = F_{k/2}L_{k/2}$ until we have an odd indexed Fibonacci number, and use a primitive for it.
So we now have only $k = 3\cdot2^i$ left to consider. $i = 0$ and $i=1$ are easy to deal with:
$F_{n + 12} - F_n = F_{6}L_{n + 6} = 8L_{n+6}$ and we know $L_{n+6} = 2x^2$ only if $n=0$ or negative (from J. Cohn's paper "Square Fibonacci Numbers, Etc." Fibonacci Quarterly 2 1964, pp. 109-113).
$F_{n + 24} - F_n = F_{12}L_{n + 12} = 12^2L_{n+6}$ and we know $L_{n+12} = x^2$ has no solutions (again barring negative Fibonacci numbers - in these cases no solutions are distinct from the positive ones).
$i=2$ causes me some trouble, and led to the negative solution $F_{36} - F_{-12} = 3864^2$. I also leave this unresolved.

For $i > 2$, we proceed as in the argument for powers of $2$. Both $7$ and $47$ divide $F_{3\cdot2^{i+1}}$ exactly once, with the base case being $F_{48}$, and they cannot both divide $L_{n + 3\cdot2^{i+1}}$, since $47 | L_m$ iff $m \equiv 8 \pmod{16}$.

The other even differences should fall the same way, although the formula for those fixes a Lucas number and varies the Fibonacci numbers, we can still find a primitive for the odd part of the Fibonacci index, but I'll have to patch up some pieces where the odd part is 3 or 1.

I've had some success with the odd differences using J. Cohn's trick: $L_m | (F_{n+2m} + F_n)$ when $3 \nmid m$ and $2|m$, and the fact that $L_m \equiv 3 \pmod{4}$ for such $m$, but no arguments covering infinitely many differences.

**EDIT:** The other even numbers are easier. Writing $F_{n + 4k+2} - F_n = L_{2k+1}F_{n + 2k+1}$ and considering $F_{n + 2k+1}$, the primitive argument removes all indices $n+2k+1$ with an odd part greater than $3$. The same argument as above works for all powers of $2$, since $3$ never divides $L_{2k+1}$. Finally, when the odd part is $3$, it's easier than before, since $7$ divides $F_{3\cdot2^i}$ for $i > 2$, $7^2$ never does, and $7$ and cannot divide the Lucas numbers since they have odd index. $i= 0, 1$ and $2$ are solved by finding all Lucas numbers which are squares, or $2$ times a square.

**EDIT 2:** For $m=2^3$, we want to know if $F_4L_{n+4} = 3L_{n+4}$ is a square. This is solved in M. Goldman. "On Lucas Numbers of the Form $px^2$ where $p=3,7,47,2207$". Math. Reports Canada Acad. Sci. (June 1988). The only example is n+4 = 2, which either does or doesn't happen according to your taste.

16more comments