No, for two reasons.
There are a lot of significant criticisms about the way mental health is addressed. It all seems very subjective, which is not good at all when you're talking about people's rights.
Even if we can judge mental health well enough to grant and take away people's rights, ultimately, it will be up to the people doing the testing whether or not they are honest with the testing and who keeps their gun rights and who doesn't. With mental health being subjective, it would be very easy to abuse the system.
If there are to be any prerequisites to gun ownership or any other rights, I believe there needs to be very straightforward and objective qualifications that are agreed on by people on both sides of the debate. That's the only way I can see it working without becoming rife with abuse.
On a side note, this is pretty much the difference between "shall issue" and "may issue" concealed carry. "Shall issue" states have to grant a concealed carry permit to anyone that can pass their tests (usually a thorough background check and a test of competence with a firearm), while "may issue" states require you to pass their test to be considered for a concealed carry permit. From what I've heard, "may issue" states require you to submit a reason for wanting a permit to the local sheriff or another governing body. So "may issue" requires a reason why you should be allowed to exercise your 2nd amendment rights, while "shall issue" requires a reason why you shouldn't be allowed those rights.
What you often hear of in "may issue" states is that "self defense" or "I live in a bad neighborhood" aren't generally accepted as good reasons. The only people with concealed carry permits in these states are politicians or people with connections to said sheriff or governing body. What qualifies as a good reason is completely up to those in charge, just like it would be for what qualifies as mentally healthy.
Again, I don't support this because it's very much open to abuse and favoritism. I prefer "shall issue" because it's objective in it's methods. If you can handle a gun and you have no criminal ties/history, there's no reason to deny you your right to carry.
Though it's important to note that it's debatable whether or not you do have a right to carry, but that's getting away from the point, which is objectivity vs subjectivity.