Answer: 9 different remainders (0, 1, 2, 3, 4, 5, 6, 7, 16)

$n$	$n^2$	$n+4$	remainder
1	1	5	1
2	4	6	4
3	9	7	2
4	16	8	0
5	25	9	7
6	36	10	6
7	49	11	5
...			...
Not much pattern so far
...			...
20	400	24	24$\times$15 = 240 + 120 = 360 24$\times$16 = 360 + 24 = 384 remainder is 16
100	10000	104	104$\times$100 = 10400 104$\times$96 = 10400 - 4016                = 10000 - 16 remainder is 16
101	10201	105	105$\times$100 = 10500 105$\times$97 = 10500 - 315               = 10200 - 16 remainder is 16

Remainder is 16 for the larger numbers tested

Had to multiply by $n-4$ for the larger values of $n$, try this algebraically: $(n-4)(n+4)=n^2-16$

So $n^2  = \underbrace{(n-4)(n+4)}_{\text{multiple of }n+4} + 16$

So the remainder is the same as the remainder when $16$ is divided by $n+4$

This will be $16$ if $n+4\gt16$ so if $n\gt12$.

Check values for $n=8$ to $n=12$ ($n$ up to $7$ shown above)

$n$	$n+4$	remainder when $16$ divided by $n+4$
8	12	4
9	13	3
10	14	2
11	15	1
12	16	0

So possible remainders are 0 - 7 and 16, 9 possibilites.