Is it permissible for one of the partners who has not paid his agreed-upon share to withdraw from the verbal partnership without paying what he owes, claiming the verbal agreement is illegitimate, and is he entitled to sell his share for an amount equivalent to what he owes?

The conditions for a capital partnership are that the capital must be present, not a debt. If a partner does not contribute anything, he does not enter into the partnership and is not entitled to any share of the profit. Rather, he is only entitled to wages for his work if he performs any. The partnership that was formed between you is invalid, and the profit is divided according to the capital contributions. Whoever did not contribute anything is only entitled to wages for his work. The absence of a written contract does not change anything, as a verbal agreement is sufficient.