No and no.
The definition of a "first-time buyer" for the purposes of the First Homes program is given by paragraph 6 of schedule 6ZA of the Finance Act, which provides that:
In this Schedule “first-time buyer” means an individual who—
(a) has not previously been a purchaser in relation to a land transaction the main subject-matter of which was a major interest in a dwelling [...].
A "major interest" is defined by Section 117 of the Finance Act to include an "an estate in fee simple absolute or a term of years absolute", subject to a few limiting statements that are not relevant here, without any limiting provision about how much of that interest is required to count as major. Accordingly, the common interpretation is that no matter how small it is, an equity share of a property counts as a "major interest" if it's a share of a freehold or leasehold.
Under the law as it stands now, there is no way to become eligible once someone has acquired a major interest. It does not matter if that person then sells their share, because an individual is eligible only if that individual "has not previously been a purchaser [...]" (emphasis added).