However, he did not believe that it was wrong for someone to own property, engage in capitalism, or anything else.

Jesus fucking Christ. He didn't believe it was wrong for someone to own property, you're correct. Because Stirner believed in "egotistical property" instead of "civic property" civic property is something that must be respected (Like private property) egotistical property is what you use, like he says, "I do not want to recognize or respect in you any thing, neither the proprietor nor the ragamuffin, nor even the man, but to use you. In salt I find that it makes food palatable to me, therefore I dissolve it; in the fish I recognize an aliment, therefore I eat it; in you I discover the gift of making my life agreeable, therefore I choose you as a companion. Or, in salt I study crystallization, in the fish animality, in you men, etc. But to me you are only what you are for me — to wit, my object; and, because my object, therefore my property."

You appear to have a small understanding of what a Union of Egoists is in your later sentences.

A union can engage in capitalistic practice (or socialistic practice). In neither case can the parties to the union be morally committed to capitalism or socialism.

Engaging in capitalistic practice is not living under conscious egoism.