In many prior activations at some point after the rules became active some broken miner managed to mine a rule violating block and some other miners who weren't broken but were just not updated yet produced blocks that continued after the invalid one.

Everyone who is updated will just ignore that block like it never happened, including all the upgraded miners.  So the invalid block and its descendants will all become stale blocks.

If you aren't upgraded you may see confirmations that go away during such an events, and if you're a miner and not upgraded you will lose money by mining on a doomed fork during such an event.

If *many* people don't upgrade such an instance could be disruptive to the ecosystem, with various services accepting transactions on the invalid fork that will be lost when the valid for overtakes it.

If for some reason you can't upgrade before activation it would be best to make sure your node(s) connect to the public network exclusively via an upgraded node so the upgraded node will act as a bad block filter.  This works for mining too, since older standard software won't introduce invalidity, only follow it if someone else introduces it.

If you can't even do that but are actively accepting payments, then you should require more confirmations before considering a transaction irreversible.  An extra ten might be a good starting point, depending on how quickly you'd respond do an incident.

If you're not accepting payments it's less critical to upgrade, but its still good to do so that your node can participate in helping the propagation of valid blocks and not propagating invalid blocks.