Find $a,b,c,d,e$ such that $\frac{s}a+1,\frac{s}b+1,\frac{s}c+1,\frac{s}d+1,\frac{s}e+1$ are all perfect squares $ (s=abcde)$ Are there five distinct positive integers $a,b,c,d,e$ such that $\dfrac{s}a+1,\dfrac{s}b+1,\dfrac{s}c+1,\dfrac{s}d+1,\dfrac{s}e+1$ are all perfect squares ? $ (s=abcde)$
If $a=1,b=2,c=12,d=2380,s=abcd=57120,$ then $\dfrac{s}a+1,\dfrac{s}b+1,\dfrac{s}c+1,\dfrac{s}d+1$ are all perfect squares. But the case of  five numbers is more difficult than four numbers.

Small contribution: if you drop the requirement that $e$ must be distinct from $a,b,c,d$ (but leave the rest the same), then the only tuples that I could find by brute force with $1+abcd\leq (2\times10^4)^2$ and some fairly large limit on $e$ are
$$
\begin{array}{ccc}
\{2,6,12,35,2\},
&\{1,2,12,1190,2\},
&\{2,4,68,105,2\},
\\\{2,10,87,156,2\},
&\{2,22,86,195,2\},
&\{2,23,90,204,2\},
\\\{2,6,247,330,2\},
&\{2,30,99,238,2\},
&\{2,58,129,360,2\},
\\\{2,52,161,396,2\},
&\{2,10,714,893,2\},
&\{2,35,1518,2014,2\},
\\\{2,12,3185,3588,2\}.
\end{array}
$$
Naturally, this doesn't answer your question.
