The relevant meaning of still is:
even; in addition; yet (used to emphasize a comparative)
"Will he not grow stronger still?" means "will he not grow stronger than he is now?" It is comparing how strong the boy already is with how strong he will grow if they don't do something to stop him - there is an implication that the boy is already stronger than they want him to be. The boy is already defying their authority - if they don't do something about it he will defy them even more.
"Will he not grow stronger?" has essentially the same meaning, but I think it loses the implication about how the speaker feels about it.
You also asked about the way you phrased this sentence in your question:
Or maybe the line still could/would be correct without it?
Would is the appropriate word to use. Could implies the possibility that it might still be correct, whereas would directly asks about the validity of such a change. (If my phrasing there sounds a little odd it's because I was trying to avoid using would in my explanation.)
I think most native speakers would find it more natural to say "would still be" rather than "still would be".