How can I get the rest of the team to commit to this?
You have to incentivize them. Talk to the CEO about a strategy for incentivizing the team. It could be money, it could be time off, it could be food, it could be team recognition, it could be tickets to a local event, etc. You know the team, you should know the best way to motivate/incentivize them. Importantly, you also must check into local labor laws to see what is allowed, as this could limit your options. For a team as small as yours, this should be the easy part and shouldn't affect the company bottom-line too much. If it does, you have deeper issues with your product that you need to be worrying about.
The next thing to do is to clearly communicate to the team that this is expected of them. They need to be able to be contacted on nights and weekends in case something happens. In other words, you need a disaster recovery plan that is clearly communicated to the team. This plan should detail expectations, how they are to be contacted, for example. If that requires a cell phone, what do you do if someone doesn't have one? Does the company subsidize everyone's phone?
Does my boss have the right to expect them to be on unpaid standby?
I don't know about a legal right, so I won't approach it from that standpoint. But, if he has never communicated to them the expectations, then no, he doesn't have the "right" to expect them to read his mind.
If he has clearly communicated to them the needs of the company and that it may occasionally require responding to a disaster, then yes, he does have the right to expect them to be on unpaid standby (unless labor law dictates otherwise). That said, a good boss would make up for it as described earlier, otherwise those who can and don't like it will leave. Those who don't like it, but can't leave (i.e., can't find a new job) will stay. You probably don't want your team filled with people like that.
To really answer this does depend on what you mean by standby. Do you mean "must be able to log in (possibly remotely) within 5 minutes of a problem"? Or do you mean "need to be available within an hour or two when a disaster strikes"? If the former, I can imagine employees being very upset and leaving without compensation. You would basically be telling them they cannot do anything on the weekends (or 1 weekend a month if this is divided among the team). If it is the latter, and it doesn't happen but twice a year, a simple paid lunch/dinner or gift cards for a night out with a significant other would probably do.