Lightning is a much worse beast than segwit - with segwit, no network effect is needed, people can set up segwit wallets and interoperate with the existing network without problems (except the new-style segwit addresses are not supported by all wallets).
I must admit I don't have full overview of lightning, but it's my understanding that bitcoins locked in the lightning network can only be sent to other participants in the lightning network (or one can try to close the lightning channel, but that's a slow process), and if a person has a wallet that doesn't support lightning, then of course he can neither receive nor send lightning funds.
From what I understand you need to setup a watcher node to make sure there is no double spending going on, if you spot a double spend then a same amount of bitcoins that where locked into the channel become yours.
As all this is happening off chain the risk of double spends is very high as the transactions only get written to the blockchain once the channel is closed.