How to determine what numbers are perfect squares without calculator? Please explain how you can do this?

This is not a complete answer, but I doubt that there is a complete answer.  For example I would not expect any easy way to tell whether or not $990297961$ is a square.
However . . . I would start by looking at the remainders when your numbers are reduced modulo $8,9$ and $5$, and seeing whether they match the possible squares in those moduli.
$$\matrix{
  &\hbox{mod $8$}&\hbox{mod $9$}&\hbox{mod $5$}\cr
  \hbox{possible squares}&0,1,4&0,1,4,7&0,1,4\cr
  23921881&1&7&1\cr
  5312660544&0&0&4\cr
  445382688&0&\color{red}{3}\cr
  1407525289&\color{red}{3}\cr}$$
The red entries above show that your third and fourth examples are not squares; the first two are still unknown.  You could try doing something for further moduli, perhaps $11$ and $7$.  But while this may rule out some numbers, it won't (at least not easily) show you that something is a square.
