Eggs
Yes, they are mystery snail eggs. Mysteries are not asexual snails, you need a male and a female, so at least one of yours is a female. The short answer to your question about saving them (presumably to hatch) is "you probably can't".
Google mystery snail eggs and you'll find plenty of good advice from people who have tried to raise them - a few successful, many not - with the details and specifics of exactly what you need to start a nursery. Something you need to know is that if you do hatch them you'll be looking at 50-100 new snails. From each clutch. An active pair can produce 2 clutches each week. Hope you love snails.
If you haven't removed that egg clutch in your pictures by now and decide to do so, notice that it's crusty on the outside and should pop off pretty easily. This is the surface of the shells hardening, which doesn't automatically mean you're going to be a new snail parent. By comparison, if you try to remove a clutch and it's totally soft and mushy, that means it was laid last night. Longer time=harder on the outside. This is how you tell how old they are. They lay at night, so look for eggs first thing in the morning.
They're always laid above the water line, at the top of the tank or underside of the lid, so this soft vs. crusty feel will be your clue as to how old it is, since you may not always immediately spot them way up there.
At a certain point it may begin to rot instead of mature, as getting a clutch to actually hatch can be tricky (see the bit about doing research on google, above). And rotting eggs aren't healthy for your Betta and his water. On the other hand, if you have a tank of goldfish, they find mystery snail eggs to be a delicious snack.
