We want to look at (integer) solutions to $$1!+2!+\cdots + x! = y^2.$$ (For clarity, if $x=1$ the left side is just $1!$, if $x=2$ it is $1!+2!$, etc.)

Examine the left side first, and see if you notice a pattern with the units digit. This should help narrow down the values of $x$ you need to examine.