[**EDITED** *to fix typos, show smoothness of $V_c$,
and extend the exhaustive-search result*]

Solutions of $P(i) = y_i^2$ ($0 \leq i \leq c$)
with $P(X) = \sum_{j=0}^3 a_j X^j$ are parametrized up to scaling by
a threefold $V_c$ in projective space ${\bf P}^c$ that's
the complete intersection of $c-3$ quadrics
$$
y_i^2 - 4 y_{i+1}^2 + 6 y_{i+2}^2 - 4 y_{i+3}^2 + y_{i+4}^2 = 0
\phantom{0\infty}
(0 \leq i \leq c-4),
$$
minus the points on the hyperplane $y_0^2 - 3 y_1^2 + 3 y_2^2 - y_3^2 = 0$
where the leading coefficient $a_3$ vanishes. We expect plenty of points
for $c < 7$, a sparse but still infinite set of points for $c = 7$,
and only finitely many points for $c > 7$ except possibly on a
proper subvariety. This last part is a special case of the Bombieri-Lang
conjecture, and if we assume this conjecture for $V_8$ then we can probably
use the forgetful maps $V_c \rightarrow V_8$ for $c>8$ to prove
(more directly than using Caporaso-Harris-Mazur as **Michael Zieve** proposed)
that some $V_c$ has no rational points except on the hyperplane $a_3=0$.

[**EDIT** René's comment raises the question of whether this
complete intersection is smooth. The answer is yes in characteristic zero.
Any linear combination of the differentials of the quadrics
$y_i^2 - 4 y_{i+1}^2 + 6 y_{i+2}^2 - 4 y_{i+3}^2 + y_{i-4}^2$
has the form $(a_0 y_0, a_1 y_1, \ldots, a_c y_c)$ with
$\sum_{m=0}^c Q(m) a_m = 0$ for any polynomial $Q(m)$ of degree at most $3$.
Therefore in any nonzero combination at least $5$ of the $a_m$ do not vanish.
Therefore at a singularity at least $5$ of the $y_i$ must be zero.
But this is impossible because $y_i^2$ are valuees of a cubic polynomial
at distinct points $i=0,1,\ldots,c$, and at most $3$ of those can be zero
unless the polynomial vanishes identically.]

An exhaustive search for rational points on $V_7$ with $a_3 \neq 0$ and
$0 \leq y_2,y_3,y_4,y_5 < 1024$ finds only the following $22$
[**EDIT** extended from $1024 = 2^{10}$ to $1536 = 3 \cdot 2^9$,
and found eight more solutions, for a total of $30$],
up to the symmetry $y_i \leftrightarrow y_{7-i}$:

```
13 7 1 1 5 7 7 1
53 21 7 29 45 53 49 3
1586 847 24 73 610 861 868 221
139 23 31 115 173 209 217 181
1061 577 35 73 469 721 883 935
31 52 47 34 35 64 107 158
821 433 49 127 355 479 473 79
139 83 71 97 125 139 127 41
359 19 79 299 439 509 481 229
163 124 107 110 121 128 121 82
169 157 119 67 55 131 233 349
368 247 134 35 76 163 242 311
826 481 164 1 286 451 544 539
323 223 167 167 197 223 223 167
595 379 187 17 109 205 251 217
973 109 239 817 1195 1387 1319 679
497 323 247 283 353 397 377 203
676 499 266 121 416 799 1226 1691
34 369 332 185 138 419 784 1203
2258 1259 356 235 838 1183 1216 521
1393 927 449 229 705 1273 1873 2499
2836 1597 470 223 1016 1471 1562 925
1373 889 475 211 317 497 581 475
1179 728 581 750 977 1124 1113 794
2027 1315 673 199 499 917 1255 1487
1027 1064 749 50 343 1252 2239 3334
2573 1629 791 65 507 923 1121 969
961 896 817 802 925 1196 1579 2042
1297 1082 827 608 613 922 1393 1948
1144 1135 854 413 388 1099 1970 2951
```

none of these lifts to a rational point on $V_8$
(neither $P(-1)$ nor $P(8)$ is a square); possibly
$c=8$ is already small enough th make a solution impossible.