You can make that argument for any pro-capitalist agenda, and it's a favorite scare-tactic to support trickle-down economics, union-busting corporations, and more. I will boycott, and urge others to do the same.
If you lie down with dogs, don't whine if you get fleas. Franchisees know exactly who they are buying into. The name comes with history, policies and politics.
Yes, some boycotts can lead to financial losses, lost jobs, etc. That's the whole bloody point. When a bad company goes down, other companies get fresh opportunities, and in turn there are fresh opportunities for employees as well. If a Papa John's goes down, customers go elsewhere to buy pizza, and they'll be hiring as demand rises.