Mercifully this never happened when I was a rep, although we did get the odd scab. What I used to do under those circumstances was... OK, rephrase that, the only one of the number of things that I used to do under those circumstances that would be appropriate with the scenario in the OP was wait for them to approach me for representation.
Obviously, I couldn't turn them down, but what I used to say was something along the lines of, "You do realise that when it comes down to it, the only power we have to stop the management from walking all over us is when we take action together? Even your 'personal case', which you think is just about you, relies on terms & conditions of employment which were won through people like us sticking together. So just you think on." Etc.
On the other hand, there's lots of things you can do to make going on strike more fun for people who don't necessarily see it as a not-to-be-missed opportunity for the articulation of working class power (i.e. most people). What we used to do was organise a post-picket social -- maybe a cheap meal out & go to the pictures or something. Not only did it make people more likely to turn out for the picket line, it gave them something to talk about the next day at work (if it was a one day strike), and helped create some feeling of belonging to something.