Does posting a general job ad break a non-solicit clause if a former employee happens to apply?

Usually not, if the non-solicitation clause is written the way most are. Non-solicitation covenants typically prohibit actively and directly approaching or soliciting specific individuals to leave their employment — they generally are not read as banning ordinary, general recruiting activity, such as a public job posting open to anyone, simply because a former employee happens to see it and apply on their own initiative. Courts asked to interpret restrictive covenants tend to distinguish targeted solicitation from a business simply hiring in the ordinary course.

The answer can change if the clause is drafted more broadly, for example if it prohibits "hiring" the covered employees at all rather than just "soliciting" them, or if it expressly covers indirect solicitation through general advertising aimed at people known to work for the other business. In that situation, a general ad that a former employee responds to could still create exposure depending on the exact wording used. Anyone relying on this distinction, whether posting the ad or watching for a breach, should confirm which version of the language actually appears in their agreement.