After a transasction it says How do I avoid this? Is this because their account is not verified? Should I not accept payments from ineligible accounts?
strange ive never recieved such a message till now. just confirm with the buyer and talk to him , make sure everything is ok.
Contact Paypal immediately and get a better understanding of the issue from the source. Then you will know what the best decision to make is.
If their account is not verified and their address not confirmed then they can request a chargeback through PayPal. Even if you are eligible under PayPal's seller protection policy they can still do a chargeback if they pay by credit card.