However, when he broadcasts a transaction on BCH, what is to stop someone from broadcasting the same transaction on the BTC network?
Nothing. That's why we say they are separate chains. This is what hard fork means. If you've found two people willing to buy your coin, but they both want different chains and you had the opportunity to obtain some before the split happens, then what you do is, essentially, double-spending.
You've doubled your belongings.
what is to prevent someone from stealing his coins on BCH?
What's the association with SegWit? If they transfer BTC to an address, whether that's SegWit or not, they've spent the UTXO. But, this happens on the Bitcoin network; there's no relation with the Bitcoin Cash network since the split happens.