The most compelling claim made by evangelists for Roman Catholicism and Eastern Orthodoxy is the argument from antiquity. They say "We have been the church since the earliest days and after all Jesus promised that the gates of hell would not stand against His church!". From there it is a short hop to accepting Roman/Orthodox dogma because if they are The Church© then of course they also have the true doctrines. I don't find it a compelling argument for two reasons, first being the general endless schisms that have plagued both Rome and Constantinople for their entire history and second because I don't think that a reasoned reading of the Scriptures leads one to see a pope/patriarch ruled ritualistic religion as the church nor do I find the critical doctrines of both to line up with Scripture. Nevertheless a lot of people who are Roman Catholic/Eastern Orthodox cling to this as their reason for being of those faiths and some people who convert find it compelling, along with the general (and not without basis) disdain for the modern evangelical church.
In a recent "Ask Anything Live" segment, Dr. Albert Mohler responded to a question about Eastern Orthodoxy and whether a person of that faith can be a real Christian. His response was great and applies to both Eastern Orthodox and Roman Catholics alike. His basic response was that someone who holds to the teaching of Eastern Orthodoxy (and Rome) faithfully has not truly found Christ and are still dead in their tresspasses and sins. That is so incredibly un-PC in this ecumenical age. As he says, justification by faith alone is not merely a doctrine, it is the Gospel. Amen and amen to that, I wish more Christian leaders would speak that boldly for the truth especially given the conversion of "The Bible Answer Man" to Eastern Orthodoxy and the popularity of Rod Dreher in parts of the church.
As far as the "We have been around the longest" argument, Mohler reply is spot on. He says in part: "For Evangelicals...we understand a Spiritual continuity of
the church. not an organizational continuity of the church.". Exactly. Jesus didn't promise us that there would be an institution that would never fail, He promised that His church, His people would never be overcome and history has demonstrated the truth of that. Even under severe persecution, often from the very organizations we are supposed to believe are the sole legitimate manifestation of the Church and the embodiment of the promise of Matthew 16:18, the church has prevailed. You don't really need to look any farther than the persecution of the Magisterial and Radical Reformers and their predecessors by Rome to know that Rome is not the church. The church is a supernatural community made up of all believers past, present and future. It often is manifested in a local, visible body but it is not always manifested that way. In other words, the organization grows out of the spirit-infused people, not the other way around. We can have the church without an institution but the institution without the people and the Gospel can never be the church.
You can watch the entire episode below, it is quite good and his response to the Revolutionary War question, while I disagree with it, is quite well thought out. Or if you prefer you can skip right to the EO question here.