I'm entirely pro-vaccines. However, I'm simultaneously entirely in favor of people asking questions, no matter what their intentions are. Their intentions do not matter. You either know your stuff or you do not. It's always better to draw out the eg anti-vaccine types. If something is defensible, then don't be a coward about it, don't be lazy, respond with a decent answer. Put in some effort. Use it as an exercise for your own brain, your own beliefs, your own ability to discuss and debate in an (ideally) civil society. Only the anti-vaccine arguments benefit from not doing that.

The shut-down angle always comes from cowardice and or intellectual weakness. You can tell a lot about someone by whether they're terrified of a discussion. Shut-down attempts are meant to evade personal shame (and more rarely to avoid putting in the time/effort), while preserving an intellectual or moral foundation established by someone else who did put in the effort.

Nearly all shut-down attempts come from weakness. If the position is important, such as vaccines, then that weakness only encourages the skeptics even more. They can practically sense it. The right thing to do is to engage and push back with facts (and or even links, with which all the hidden passive readers can go make up their own minds as to whether they agree or not), even if you don't have all day to argue a topic.