I cannot find evidence for a third party using the public financing option for either state. I can see nothing preventing it and assume they have at some point. Specifically I'm referring to the $5 multiple donations needed that then lead the state to fully fund their campaign.
Maine's 2018 Candidate list (Excel) shows that of 211 candidates choosing to use the public financing option, 140 were Democrats, 54 Republicans, 16 Unenrolled, and 1 Green.
There are additional reports available on Maine's Governmental Ethics and Elections Practices site, but the 2018 list alone is sufficient to answer that third party candidates have used public financing in Maine.
Arizona's Historical Candidate Data page shows 18 third party candidates (Green, Libertarian, and Independent) availing themselves of the public financing option since it was enacted.