Arguments Against Ethical Egoism

The first premise is certainly false. “If we (everyone) minded her own business, and tended to their own needs, then everyone would be better off.” This premise, as it is, cannot hold up the self-reliance argument let alone ethical egoism. If John, an elderly man, was walking in the park and had a fall, would he be better off if Jill, a bystander, just continued on her way? She was running late to work and could easily tend to her own needs leaving John on the ground moaning in pain. The second premise, “We ought to do what will make everyone better off.” may be true, but it does not support ethical egoism. It is really a utilitarian claim; utilitarians think that one ought to do what will maximize the well-being for the most people. While ethical egoists think that one ought to do what will maximize their own well-being, they might care about the well-being of others to the extent that it would make themselves better off in the end.
Ethical egoism is all about doing what will make oneself better off. The interests of others by and of itself, counts for nothing to an ethical egoist. A modified version could be as follows:
Premise 1: If one tends to their own needs and minds their own business (is self-reliant) then one will be better off (their well-being is