l want to know if we have scriptural authority which forbids children and non-believers to partake the Lord's Supper?
I know of no passage that places a requirement on who may partake of the Lord's Supper. Partaking of the Lord's Supper is commanded of Christians (I Corinthians 10:16-17). It has meaning to the Christian because it is a memorial of Jesus' death. "For as often as you eat this bread and drink this cup, you proclaim the Lord's death till He comes" (I Corinthians 11:26). It doesn't hold the same meaning for the non-Christian. But so long as it is being taken in a respectful manner, worthy of Christ's death (I Corinthians 11:27), who partakes is not restricted.
Notice that "But let a man examine himself, and so let him eat of the bread and drink of the cup" (I Corinthians 11:28) tells the individual to examine himself as to why he is partaking. This is not a command to those offering the Lord's Supper to examine those who are wanting to partake to determine if they are worthy.
As parents, when our children were small, we did not allow our children to partake because we wanted them to see that this was something special for Christians. Other parents may have other ideas. But when someone is a non-Christian, whether they eat or not does not make any difference.