There is no such thing as "no downtime, ever." The sooner your boss understands that, the more rational discussion you can have.
There will always be downtime, but you want to minimize it. If you use log shipping, you will have some delay in terms of how often you back up the logs and move them. That can affect your RPO in terms of a bit of data loss. It also affects your RTO, since you need to move clients to the new instance/database, run the new database through recovery (usually quick), and deal with any networking issues. If you are trying to protect yourself from data loss and local server loss, log shipping is great. It's simple, and more importantly, if you need to move the logs and backup to a new server because of some issue with your secondary site, you can do it.
Mirroring gets closer to no downtime, and no data loss, but it's more complex to set up, and you'd have to make sure you have a secure pipe and connection between the two instances. Your clients (if they are relatively new), can make the switch automatically to the new server/database without networking issues.
Replication can provide no downtime if the clients were to know to switch to the new server, but that can be complex. You also could have PK issues in trying to put new data onto the secondary server. It's the most complex, administratively, and the most likely to break. It's also the messiest in terms of the data merge, and the fail back.
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