This is a question that divides evangelicals, sometimes sharply. But perhaps it's not the question to ask. At least not first. Perhaps the better question with which to frame the discussion is this:
Where (if anywhere) might counselling fit into Christian discipleship in the life of the church?
Here we're dealing with first things first. Healthy, well-balanced living is a matter of Christian discipleship - learning Christ and letting His reality shape all our thinking, feeling and behaving. The context for that discipleship is the body of Christ - the church. Think of Ephesians 4: there is incredible maturity and growth available in the renewal of our minds (v14-32). But the context for it all is the local church (v1-13). I say local because you cannot imagine Ephesians 4 happening in some "invisible church." Our growth happens in community with others where our church leaders exercise their word gifts (v11-13). The local church is the place where lives are put back together.
So where does counselling fit in? Well it certainly can fit in. There are areas of expertise that a counsellor might have that may not be covered by your local church. There can be many reasons you might want to talk to someone with wisdom and experience from beyond your own context. But the heart of the issue is this: the counsellor must not become a de facto Pastor. Whatever counselling is sought, it would probably need to be self-consciously time-limited. It will certainly need to be talked and prayed through with people from church (whether friends or pastoral leaders). But if that's happening then seeing a non-Christian counsellor for a limited time, regarding a limited issue and with local church prayer support is a viable option - certainly it is if your Pastor thinks so.
The real problem, it seems to me, is having a counsellee looking to a counsellor as a Pastor - a shepherd of their soul. Unfortunately though, this happens all the time. And the problem is not limited to counsellors. So many want their soul's deepest needs to be met through celebrity online preachers, through parachurch ministries, through conferences, through GOD TV, through the latest Christian paperback. This is the problem, and you'll notice it's a problem even if the spiritual shepherd being sought is Christian. Even if their doctrine is completely orthodox, the problem remains.
On the other hand, consider this scenario. Let's imagine the person you love is falling into a big black hole called anorexia - this is Emma's and my story. In that case there will be professional help available - not always very much, not always very helpful and not always very easy to access. But nonetheless there will be opportunities to access care from beyond the local church and, most often, beyond an explicitly Christian context. What should you do?
My advice: take everything that's offered with both hands, but pray through it with folks from church. Christ and His people really do have the answer to eating disorders but there's also wisdom out there about the nuts and bolts of battling it. Through it all, church should absolutely maintain spiritual oversight and care but church should also acknowledge where others have useful expertise. The problem comes when Christian leaders equate expertise with soul care. In that case leaders who feel "out of their depth" in terms of expertise will hand over their people wholesale to the experts.
This happened to Emma and I several times. We would go to Christian leaders with our problems only to be told they were "too much" for them to handle. This was understandable - they were "too much" for us too! But instead of maintaining soul care while delegating specific expertise, they washed their hands of both.
The greatest help we had, beyond the Lord's own intervention, was a Christian couple who admitted frequently their own inadequacy. They had no idea about eating disorders but every Monday night they opened up their home and we talked. Emma could speak about the professional help she was receiving and we prayed together. Very simple. Very profound. And it was all possible because these Christian leaders did not consider their limited knowledge to limit their ability to care.
Can non-Christian counsellors be helpful? Of course. Can they be unhelpful. You bet. But the need to find "an answer" to the counselling question betrays a deeper problem - we feel like the counsellor, whether Christian or not, is the answer. We treat outside helpers like alternative shepherds when really they're just vets.
For my money the real issue is dethroning the counsellor from the position of Pastor. Once this occurs, their doctrinal orthodoxy is still important, but less so. You see shepherds can bring in vets to help the flock. And those vets may be a brother, or they may be from a different tribe. That's of secondary importance. Whoever the vet is, the real issue is that the sheep know the voice of their Shepherd and the continued care of their God-given undershepherds.